Discriminants of Hermitian R[G]-modules and Brauer’s Class Number Relation

Ernst Kani*

Abstract. The purpose of this paper is to lay the foundations for a quantitative theory of relations among discriminants of hermitian RG-modules which are induced by character relations. This is accomplished by introducing an invariant δ(M) attached to an RG-module M which plays the role of a correction term in such relations and to study its functorial properties such as localization and induction theorems, behaviour with respect to exact sequences, triviality etc. By means of this formalism it is shown that this invariant may be computed in many cases.

An application of this invariant is the class number relation of R. Brauer (1951) and, by using the formalism mentioned above, also that of Dirichlet (1842).

1. Introduction

A hundred and fifty years ago Dirichlet[Di2] proved the following theorem which he considered to be “one of the most beautiful theorems of the theory of complex numbers” ([Di1], p. 508).

Theorem 1.1 (Dirichlet, 1842) The class number h(K) of Dirichlet’s biquadratic field K = ℚ(\sqrt{d}, \sqrt{−d}), where d > 1 is square-free, is given by the formula

h(K) = \frac{1}{2}Qh(d)h(−d). (1.1)

Here h(±d) denotes the class number of the quadratic field ℚ(\sqrt{±d}) and

Q = \{ \begin{matrix}1 & if N(ε_{K}) = ±1 \\ 2 & if N(ε_{K}) = ±i, \end{matrix} (1.2)

where ε_{K} denotes a fundamental unit of K, except in the case that K = ℚ(ζ_{8}) where ε_{K} = ζ_{8} denotes an eighth root of unity, and N(ε_{K}) = N_{K/ℚ(i)}(ε_{K}) denotes its “partial norm”.

*NSERC University Research Fellow

Dirichlet’s own proof of this theorem was by analytic techniques, but in 1896 Hilbert[Hi] gave an algebraic proof of this result. In 1922 Herglotz[He] extended this theorem to arbitrary biquadratic fields, and this was further generalized by Kuroda[Ku] in 1950 (see also Kubota[Kb] and Walter[Wa2]).

For a general Galois number field, Nehrkorn[Neh] obtained in 1933 some interesting partial results. The first significant progress, however, was achieved in 1951 by R. Brauer[Br] who proved the following theorem.

Theorem 1.2 (Brauer, 1951) Let K be a Galois extension of ℚ with Galois group G. Suppose G is non-cyclic, so that we have a non-trivial character relation

∑_{H≤G} n_{H}1_{H}^{∗} = 0, (1.3)

where the sum extends over all subgroups of G and 1_{H}^{∗} denotes the permutation character attached to G/H. Then the class numbers h(K^{H}) of the fixed fields K^{H}, H ≤ G, satisfy the relation

∏_{H} h(K^{H})^{n_{H}} = ∏_{H}(w(K^{H})[G : H])^{n_{H}} · J = ∏_{H}(w_{2}(K^{H})[G : H])^{n_{H}} · J, (1.4)

where w(L) = |U(L)_{tor}| denotes the number of roots of unity in the field L and w_{2}(L) = |U(L)_{tor}^{(2)}| the number of those of 2-power order, and J is an “invariant” which depends only on the ℤ[G]-module structure of the group of units U(K) of K (and on {n_{H}}).

By using the Jordan–Zassenhaus Theorem that there are up to isomorphism only a finite number of ℤG-lattices of bounded rank, Brauer derived the following corollary which may be viewed as a qualitative generalization of Dirichlet’s theorem:

Corollary 1.3 Fix a group G and a character relation (1.3). Then, as K runs over all Galois extensions of ℚ with group G, the product

∏_{H} h(K^{H})^{n_{H}}

assumes only finitely many values.

Brauer’s proof consists of two main steps. In the first he uses the Artin formalism to derive from the character relation (1.3) the relation

∏_{H} ζ_{K^{H}} (s)^{n_{H}} = 1 (1.5)

among the Dedekind zeta-functions, and from this he obtains the relation

∏_{H} h(K^{H})^{n_{H}} = ∏_{H}(w(K^{H})reg(K^{H})^{−1})^{n_{H}} (1.6)

by comparing residues at s = 1. Here, reg(L) denotes the regulator of L.

His second step consists of a detailed study of the right hand side of (1.6), and culminates in the proof of the regulator relation

J = ∏_{H}([G : H]reg(K^{H}))^{−n_{H}} , (1.7)

where J is a certain invariant of the ℤG-module U(K) which is defined as the product of suitable indices of unit groups. Brauer’s exposition is somewhat difficult to follow in this step, but a much more transparent treatment with considerable simplifications was given by C. Walter[Wa1] in 1979. In particular, Walter clarified Brauer’s use of the Minkowski units by showing that they play the role of a “comparison sublattice”.

In this paper we shall follow in the second step a different approach, one that was in part inspired by Arakelov theory for number fields (cf. Neukirch[Neu] or Szpiro[Sz]). The underlying idea here is that the regulator of a number field K may be viewed as the discriminant of the group of units U(K) with respect to a suitable bilinear form (or metric), a fact that is implicit in the Arakelovian point of view of number theory. In the case that K/ℚ is Galois with group G, then M = U(K) is a ℤ[G]-module with a G-equivariant metric ρ; following the usage of Scharlau[Sch], such a pair (M, ρ) will be called a hermitian ℤ[G]-module.

Viewed in this light, it is natural to expect that Brauer’s theorem may be deduced from a very general theorem on relations among discriminants of hermitian ℤG-modules, and this turns out to be indeed the case.

The theorem in question is based on the observation that if {n_{H}} defines a character relation (1.3), then the expression

δ^{∗}(M) = ∏_{H} disc(M^{H}, h_{|M^{H}})^{n_{H}} (1.8)

does not depend on the choice of the hermitian structure h on the ℤG-module M and hence defines an invariant δ^{∗}(M) = δ^{∗}({n_{H}}, M) of the ℤG-module M. On the other hand, we can also view equation (1.8) as a discriminant relation, one that includes Brauer’s regulator relation (1.7) as a special case, once we have interpreted regulators as discriminants (cf. Theorem 2.7).

Here it should be emphasized that the invariant δ^{∗}(M), although closely related to Brauer’s invariant J, is nevertheless substantially different from that of Brauer since its definition does not require the existence of a “comparison sublattice”. As a result, the entire theory becomes not only more flexible and more functorial but also much more general in that the general discriminant theorem (Theorem 2.5) also applies to hermitian ℤ[G]-modules which have no obvious comparison submodules. A typical example here is the Mordell–Weil lattice of an elliptic curve for which one can therefore obtain an analogue of Brauer’s theorem, as will be shown in a subsequent paper.

While the above discriminant relation (1.8) suffices to prove Brauer’s class number relation and its corollary, it sheds little light on the value of the invariant δ^{∗}(M). The main purpose of this paper, therefore, is to introduce a formalism with which the invariant δ(M) may be computed in many cases and thereby to lay the foundations for a quantitative theory of discriminant relations induced by character relations. This is accomplished by studying some functorial properties of the δ-invariant such as localization, reciprocity theorems, behaviour with respect to exact sequences etc. In particular, it will be shown how to deduce Dirichlet’s theorem directly by examining the ℤ[G]-module structure of U(K) (cf. Proposition 2.18d)).

In developing this formalism, it is advantageous to modify the above invariant somewhat. First of all, since the unit groups invariably involve torsion — indeed, the case distinction in Dirichlet’s Theorem is precisely due to the fact that the ℤG-module structure on U(K) cannot be read off from that of its torsion-free part \overline{U(K)} = U(K)/U(K)_{tor} — it is useful to redefine the definition of a discriminant in such a way as to be sensitive to its torsion subgroup. Moreover, by introducing these modified discriminants, many proofs actually become simpler since these new discriminants satisfy better functorial properties.

The second modification is to introduce a normalization factor when restricting a hermitian pairing h on M to its invariant submodule M^{H} (cf. Notation 2.4). Although the resulting modified invariant δ(M) is just a minor variation of the original one (cf. Remark 2.6 for the precise connection), this modification is absolutely essential for the validity of many of the functorial properties such as Frobenius reciprocity (Theorem 2.12) and the Triviality Theorem 2.9. Moreover, it also turns out to be the natural choice for the hermitian pairings on unit groups (cf. Proposition 8.2).

The basic method for proving the discriminant relation (1.8) and the other results is to translate the character relation (1.3) into an isomorphism

M^{+} ⊗ ℚ ≃ M^{−} ⊗ ℚ (1.9)

of ℚG-modules, where M^{+} and M^{−} are certain permutation modules attached to the character relation (1.3). Viewed in this way, the entire theory can generalized as follows.

Given a non-degenerate ℤG-module M and two ℤG-modules M_{1} and M_{2} which are symmetric (or self-dual), i.e. M_{i}^{∗} ≃ M_{i}, and which satisfy (1.9), the expression

δ(M_{1}, M_{2}; M) = d_{G}(M_{1} ⊗ M, h_{1} ⊗ h)d_{G}(M_{2} ⊗ M, h_{2} ⊗ h)^{−1} (1.10)

does not depend on hermitian structure h on M nor on the unimodular hermitian structures h_{i} on M_{i}, and hence defines an invariant of M; here,

d_{G}(M, h) = disc(M^{G}, \frac{1}{|G|}h_{|M^{G}}).

Moreover, virtually all the functorial properties of the invariant δ^{∗}({n_{H}}, M) can be derived from the analogous properties of the invariant δ(M_{1}, M_{2}; M), as will be shown in sections 6 and 7.

This paper is organized as follows.

For the convenience of the reader we present in section 2 an overview of the main definitions and results of the paper. In this overview we focus only on the results for the invariant δ({n_{H}}, M), but it should be emphasized that virtually all the theorems here are special cases of the theorems of section 6 which apply more generally to the invariant δ(M_{1}, M_{2}; M).

In section 3 we define and study the functorial properties of the (modified) discriminant of a bilinear R-module. For this purpose we first introduce the content ideal χ(f) attached to a linear map f : M → N, which extends the notion of the content ideal χ(T) of a torsion module (cf. Bourbaki[BCA]) and also that of the “Herbrand quotient” of Fulton[Fu], Appendix A.2, and which reduces to the “qinvariant” of Tate[Ta] in the case that R = ℤ. This will then be used to define the relative invariant χ(f, g) of two R-linear maps f, g : M → N which refines and generalizes the relative invariant of Bourbaki attached to submodules. Finally, the discriminant will be defined as a suitable relative invariant.

In section 4 we define hermitian RG-modules and study their functoriality properties via the process of (co)induction and restriction. In particular, we shall formulate several versions of a generalized Frobenius reciprocity theorem for hermitian RG-modules. While the presentation here is much more elaborate than is necessary for our purposes, it should be viewed as a contribution towards an equivariant Arakelov theory.

In section 5 we construct the fundamental invariant δ(M_{1}, M_{2}; M). By introducing suitable Grothendieck rings, it is possible to view this invariant as a pairing and thereby work with it more efficiently.

In section 6 we formulate and prove the main properties of the invariant δ(M_{1}, M_{2}; M). As was mentioned above, most of these properties are natural generalizations of the properties of the invariant δ({n_{H}}, M) which are presented in section 2: the theorem on discriminant relations, the Uniform Boundedness Theorem, the theorems on induction and inflation, behaviour with respect to exact sequences, and the Triviality Theorem.

In section 7 we interpret the results of section 6 in terms of character relations and thereby prove the theorems stated in section 2.

Finally, in section 8 we apply the above theory to study relations among Sregulators and S-class numbers of number fields by calculating the δ-invariant of the group U_{S}(K) of S-units. As an application, we prove Dirichlet’s Theorem and a generalization of Brauer’s Theorem.

This paper developed out of the joint work [KR2] with M. Rosen, whom I would like to thank very much for the many fruitful discussions as well as for his interest and encouragement throughout the project. In addition, I have also benefitted from discussions with J. Ritter and A. Weiss. Finally, I would like to gratefully acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC).

2. Main results

Throughout this paper, the following notations and assumptions are in effect. Let

R be a principal ideal domain, K = Quot(R) denote its quotient field, L ⊃ K be an extension field, G a finite group with char(K) |̸ n = |G|, RG = R[G] the group ring of G with coefficients in R.

Actually, virtually all of the results below are valid if R is a Dedekind domain or even an arbitrary integrally closed noetherian domain, but for simplicity we restrict to the case of a principal ideal domain since it is the most general ring required for the applications which use mainly the case R = ℤ and L = ℝ.

The following definitions are fundamental for the entire paper and will be discussed in more detail in the next sections.

Definition 2.1 An L-valued hermitian RG-module is a pair (M, h), where M is a finitely generated RG-module and

h : M → M^{∗} ⊗ L = Hom_{R}(M, L)

is an RG-linear map. Here M^{∗} = Hom_{R}(M, R) denotes the contragredient RGmodule. We call (M, h) non-degenerate if the induced map

h ⊗ L : M ⊗ L → M^{∗} ⊗ L

is an isomorphism.

If G = {1} is the trivial group, then a hermitian RG-module (M, h) will be called a bilinear R-module.

Remark 2.2 It follows from the definition that an RG-module M which admits a non-degenerate L-valued RG-module structure h : M → M^{∗} ⊗ L satisfies the symmetry condition M ⊗ L ≃ M^{∗} ⊗ L, or equivalently , M ⊗ K ≃ M^{∗} ⊗ K, which says that the KG-module M ⊗ K is isomorphic to its contragredient module (as KG-modules). An RG-module M with this property will be called non-degenerate.

Definition 2.3 The discriminant of an L-valued bilinear R-module (M, h) is the (principal) R-submodule of L defined by

disc(M, h) = χ(M_{tor})^{−1} det(h(x_{i}, x_{j})) · R,

where x_{1}, … x_{n} ∈ M is a basis of \overline{M} = M/M_{tor}, with M_{tor} denoting the torsion submodule of M. Furthermore, h(x_{i}, x_{j}) = h(x_{i})(x_{j}), and χ(⋯) denotes the content ideal in the sense of Bourbaki [BCA]. (In particular, if R = ℤ then χ(M_{tor}) = |M_{tor}| · ℤ.)

To define the invariant δ({n_{H}}, M), we introduce the following notation.

Notation 2.4 For a hermitian RG-module (M, h) and a subgroup H ≤ G, define the bilinear R-module Inv_{H}(M, h) by

Inv_{H}(M, h) = (Inv_{H}(M), inv_{H}(h)),

where Inv_{H}(M) = M^{H} = {m ∈ M : hm = m} denotes the R-submodule of H-invariant elements and

inv_{H}(h) = \frac{1}{|H|}h^{H} : M^{H} → (M^{∗} ⊗ L)^{H} = (M^{∗})^{H} ⊗ L

denotes the induced map on the invariant spaces multiplied by the factor \frac{1}{|H|}. The reason for introducing this normalization factor will be explained below; cf. Theorems 2.9 and 2.12, and also section 4.

With these definitions and notations we have:

Theorem 2.5 (Discriminant Relation) Let {n_{H}}_{H≤G} define a character relation (1.3). Then we have:

a) (Existence) For any non-degenerate RG-module M there is a unique principal fractional R-ideal

δ(M) = δ_{R}({n_{H}}, M)

such that for any non-degenerate L-valued hermitian RG-structure on M we have:

δ(M) = ∏_{H} disc_{R}(Inv_{H}(M, h))^{n_{H}} . (2.1)

b) (Additivity) The invariant δ_{R}({n_{H}}, M) is bi-additive:

δ_{R}({n_{H}}, M_{1} ⊕ M_{2}) = δ_{R}({n_{H}}, M_{1}) · δ_{R}({n_{H}}, M_{2}),
δ_{R}({n_{H} + m_{H}}, M) = δ_{R}({n_{H}}, M) · δ_{R}({m_{H}}, M).

c) (Base change) If R′ ⊃ R is a principal ideal domain then we have:

δ_{R′}(M ⊗_{R} R′) = δ_{R}(M) · R′.

d) (Support) We have δ_{R}(M) · R[\frac{1}{n}] = R[\frac{1}{n}], so δ_{R}(M) is supported on the prime ideals dividing n = |G|.

e) (Localization) The following localization formula is valid:

δ_{R}(M) = ⋂_{𝔭|n} δ_{R_{𝔭}}(M_{𝔭}); (2.2)

here, the intersection runs over all prime ideals 𝔭 of R which contain n = |G|. In particular, if M and M′ are in the same genus (i.e. M_{𝔭} ≃ M′_{𝔭}, for all 𝔭|n), then

δ_{R}(M) = δ_{R}(M′). (2.3)

Remark 2.6 In the applications one is often interested in the (unnormalized) product

δ_{R}^{∗}({n_{X}}, M) = ∏_{X} disc_{R}(M^{H}, h_{|M^{H}})^{n_{H}} (2.4)

in place of the product (2.1) which is calculated using the normalization introduced in Notation 2.4. However, it is easy to relate this quantity to the invariant δ:

δ^{∗}({n_{H}}, M) = ε({n_{H}}, M)δ({n_{H}}, M), (2.5)

where

ε({n_{H}}, M) = ∏_{H} |H|^{n_{H}rk(M^{H})}. (2.6)

In particular, it follows that all the assertions of Theorem 2.5 remain true when we replace δ by δ^{∗}.

We note that it follows from (2.5) that the invariant of the trivial module M = R is given by

δ_{R}({n_{H}}, R) = ∏_{H} |H|^{−n_{H}} · R (2.7)

because it is immediate that δ^{∗}({n_{H}}, R) = R.

In order to be able to deduce Brauer’s results from Theorem 2.5, we need to relate the δ-invariant of the unit group U(K) to the regulators of the subfields of K. As a consequence of the formalism of δ-invariants introduced below, this can be done more generally for the group U_{S}(K) of S-units of K:

Theorem 2.7 (Class Number Relation) Let K be a Galois extension of a number field K_{0} with Galois group G, and let S be a finite set of places of K which is G-invariant and contains the set S_{∞} of infinite places. Then the invariant of the the S-unit group U_{S}(K) of K, viewed as a ℤ[G]-module, is given by

δ_{ℤ}(\{n_{H}\}, U_{S}(K)) = ∏_{H} (\frac{reg_{H\S}(K^{H})^{2}}{w(K^{H})} )^{n_{H}} δ_{ℤ}(\{n_{H}\}, ℤ)δ_{ℤ}(\{n_{H}\}, ℤ[S])^{−1},

where reg_{H\S}(K^{H}) denotes the S̄-regulator of K^{H} with respect to the set S̄ = H\S of places of K^{H}, and the δ-invariants δ_{ℤ}({n_{H}}, ℤ) and δ_{ℤ}({n_{H}}, ℤ[S]) are given by (2.7) and (2.19), respectively.

We thus obtain the following relation among the S-class numbers h_{H\S}(K^{H}) of the intermediate fields K^{H} of K/K_{0}:

∏_{H} h_{H\S}(K^{H})^{2n_{H}} = δ(ℤ)δ(U_{S}(K))^{−1}δ(U(K)_{tor})^{−1}δ(ℤ[S])^{−1}. (2.8)

Moreover, if the (finite) places of S are tamely ramified over K_{0} (e.g. if S = S_{∞}), then δ_{ℤ}({n_{H}}, ℤ[S]) = ℤ, and so the above class number relation reduces to

∏_{H} h_{H\S}(K^{H})^{2n_{H}} = δ(ℤ)δ(U_{S}(K))^{−1}δ(U(K)_{tor})^{−1}. (2.9)

If G, {n_{H}} and #S are fixed, then, as was explained in the introduction, it follows immediately from the Jordan-Zassenhaus Theorem that the expression (2.8) assumes only finitely many values as K varies over all Galois extensions of ℚ with Galois group G. However, this result also follows from the following much more general and more precise theorem:

Theorem 2.8 (Uniform Boundedness) For each character relation (1.3) there is an element r = r({n_{H}}) ∈ R\{0} such that for any non-degenerate RG-module M we have

[M_{tor}]^{N}r^{rk(M)}δ_{R}({n_{H}}, M) ⊂ R, [M_{tor}]^{N}r^{rk(M)}δ_{R}({n_{H}}, M)^{−1} ⊂ R,

where N = ∑ |n_{H}|.

Note that this theorem is considerably stronger than Brauer’s result. Not only is the bound here uniform in the rank of M, but the result is also valid for arbitrary principal ideal domains R, not only for those for which the Jordan– Zassenhaus theorem holds. Furthermore, as the proof shows, it is possible to give fairly explicit interpretation of the “denominator” r above: its essential constituent is the “genus defect” attached to a character relation, which measures the failure of the character relation to be trivial as a genus class (cf. sections 6 and 7 for further explanations).

In view of the above class number relation (2.8), it would be desirable to be able to compute the invariant δ_{R}(M) for all (non-degenerate) RG-modules M. While this task seems to be quite difficult in general, the following formalism can be used to compute this invariant in many cases.

Theorem 2.9 (Triviality) If M is a non-degenerate RG-module which is cohomologically trivial then its invariant is also trivial:

δ_{R}({n_{H}}, M) = R. (2.10)

The above theorem may be viewed as a justification for the normalization factor introduced in the definition of δ, as well as for the unusual definition of the discriminant of RG-modules M which may have R-torsion. A further justification of the latter is also given by the validity of the following result:

Theorem 2.10 (Exact Sequence Formula) Let

0 → M′ →^{f} M →^{g} M′′ → 0

be an exact sequence of RG-modules. If any two of the modules M, M′ and M′′ are non-degenerate, then so is the third, and for any character relation we have

δ({n_{H}}, M) = δ({n_{H}}, M′)δ({n_{H}}, M′′)ψ({n_{H}}, g)^{2}, (2.11) where

ψ({n_{H}}, g) = ∏_{X} χ(Coker(g^{X} : H^{0}(X, M) → H^{0}(X, M′′)))^{n_{X}}
= ∏_{X} χ(Ker(f : H^{1}(X, M′) → H^{1}(X, M)))^{n_{X}} .

Combining these two theorems yields the following remarkable consequence:

Corollary 2.11 Let

0 → M′ → P_{r} → ⋯ → P_{1} → M′′ → 0

be an exact sequence of non-degenerate RG-modules. If the P_{i} are cohomologically trivial, then for any character relation {n_{H}} the δ-invariants of M′ and M′′ are connected by the formula

δ({n_{H}}, M′) = δ({n_{H}}, M′′)^{(−1)^{r}} · ∏_{H} χ^{r}(H, M′)^{2n_{H}} , (2.12)

where the truncated Euler characteristic χ^{r}(G, M) is defined by

χ^{r}(G, M′) = ∏_{q=1}^{r} χ(H^{q}(G, M′))^{(−1)^{q+1}} (2.13)

The above corollary allows us to compute the δ-invariant of one RG-module in terms of another one. The following induction/restriction formalism, on the other hand, relates the δ-invariant of (certain) RG-modules to those of modules belonging to subgroups of G.

Theorem 2.12 (Induction) Let X ≤ G be a subgroup. If M, respectively N, is a (non-degenerate) RG-module, respectively RX-module, and {n_{H}}_{H≤G}, respectively {m_{Y} }_{Y ≤X}, defines a character relation of G, respectively of X, then we have

(2.14)  δ_{R}(ind_{X}^{G}({m_{Y} }), M) = δ({m_{Y} }, res_{X}^{G}(M)),
(2.15)  δ_{R}(res_{X}^{G}({n_{H}}), N) = δ({n_{H}}, ind_{X}^{G}(N)),

where the induced character relation ind_{X}^{G}({m_{Y} }) = {m_{H}^{∗}}_{H≤G} is defined by extension by 0 (i.e. m_{H}^{∗} = m_{H}, if H ≤ X and m_{H}^{∗} = 0 otherwise), and the restricted character relation res_{X}^{G}({n_{H}}) = {n_{Y}^{∗} }_{Y ≤X} is defined by the formula

n_{Y}^{∗} = ∑_{H} n_{H}m(H, Y) (2.16)

where m(H, Y) = #{g ∈ G : gHg^{−1} ∩ X = Y }/(|H|[X : Y]).

Note that although the above coefficients m(H, Y) of res_{H}^{G} are actually rational numbers, the denominators disappear if one collects those terms together which belong to conjugate subgroups; this is possible since the δ-invariant does not change if we replace in a character relation a subgroup by a conjugate.

The above notation suggests that Theorem 2.12 is closely connected to a “Frobenius induction theorem”. This is indeed the case, for the result is deduced from the “Frobenius induction theorem for hermitian RG-modules” (Theorem 4.16).

As an illustration of the above theorem we present the following examples. The first two follow from the fact that cyclic groups have no non-trivial character relations and that the process of restriction and induction preserves character relations.

Example 2.13 a) If M = ind_{X}^{G}(N) is induced from a cyclic subgroup X ≤ G, then for every character relation {n_{H}} we have

δ_{R}({n_{H}}, M) = R. (2.17)

b) If G acts tamely on the G-set S in the sense that all stabilizers are cyclic, then for the associated permutation module R[S] we have

δ_{R}({n_{H}}, R[S]) = R. (2.18)

c) For any G-set S we have

δ_{R}({n_{H}}, R[S]) = ∏_{s∈G\S} ∏_{H} ∏_{g∈H\G/G_{s}} |H ∩ gG_{s}g^{−1}|^{−n_{H}} · R, (2.19)

where G_{s} denotes the stabilizer of a point s ∈ S.

Another useful induction-type theorem is the following result which relates the δ-invariants of modules which are lifted (or “inflated”) from quotient groups.

Theorem 2.14 (Inflation) Let X ⊴ G be a normal subgroup of G and let Q = G/X denote the quotient group.

a) Let M be a non-degenerate RG-module and {n_{Y} }_{Y ≤Q} be a character relation of Q. Then, viewing the invariant module M^{X} as an RQ-module, we have

δ_{R}(inf_{G}^{Q}({n_{Y} }), M) = δ_{R}({n_{Y} }, M^{X}), (2.20)

where the lifted character relation inf_{G}^{Q}({n_{Y} }) = {n_{H}^{∗}} is given by n_{H}^{∗} = 0 if H ≱ X and by

n_{H}^{∗} = ∑_{Y ∼H/X} n_{Y} /[G : N_{G}(H)],

if H ≥ X; here the sum extends over all Y ≤ Q which are conjugate to H/X.

b) For every non-degenerate RQ-module M and character relation {n_{H}} of G we have δ_{R}^{∗}({n_{H}}, Res_{G}^{Q}(M)) = δ_{R}^{∗}(ind_{G}^{Q}({n_{H}}), M), (2.21)

where the induced character relation ind_{G}^{Q}(\{n_{H}\}) = \{\overline{n}_{Y/X}\}_{Y/X≤Q} is defined by

\overline{n}_{Y/X} = ∑_{{H \atop H·X=Y}} n_{H}.

In terms of the δ-invariant, equation (2.21) may be written as

δ_{R}(ind_{G}^{Q}({n_{H}}), M) = ε_{X}({n_{H}}, M)δ_{R}({n_{H}}, Res_{G}^{Q}(M)), (2.22)

where ε_{X} is defined by

(2.23)  ε_{X}({n_{H}}, M) = ε(ind_{G}^{Q}({n_{H}}), M)ε({n_{H}}, Res_{G}^{Q}(M))^{−1}
= ∏_{H} |H ∩ X|^{n_{H}rk(M^{H})}.

Actually, Theorems 2.12 and 2.14 are just two special cases of a theorem which relates the δ-invariants of modules which are restricted and/or (co)induced via an arbitrary group homomorphism f : G_{1} → G_{2}; cf. Theorem 7.8 for a precise statement.

As with restriction to subgroups, the process of inflation and quotient-induction preserves character relations. Thus, if we use once more the fact that cyclic groups have no non-trivial character relations, then we obtain from the above theorem the following examples.

Example 2.15 a) If M is an R-module on which RG acts trivially, then we have

δ_{R}({n_{H}}, M) = δ_{R}({n_{H}}, R)^{rk(M)} = ∏_{H} |H|^{−n_{H}rk(M)}R; (2.24)

this generalizes and gives another proof of (2.7).

b) If M = M^{X} for some normal subgroup X⊴G with cyclic quotient Q = G/X, then δ_{R}({n_{H}}, M) = ε_{X}({n_{H}}, M)^{−1}R. (2.25)

Corollary 2.16 If M is an RG-module which is a torsion module then

ε_{X}({n_{H}}, M) = 1,

and so equation (2.21) holds with δ^{∗} replaced by δ.

In particular, if M = M^{X} is a torsion module which is invariant under a normal subgroup X ⊴ G with a cyclic quotient G/X, then

δ_{R}({n_{H}}, M) = R. (2.26)

Remark 2.17 This corollary applies in particular to the case that M = U(K)^{(p)}, the p-primary component of the torsion subgroup of the unit group of a galois extension K/K_{0} as in Theorem 2.7, provided that p > 2. Thus, in the notation of Theorem 1.2 we obtain Brauer’s relation (i.e. the second equality in (1.4)) as a special case:

∏ w_{p}(K^{H})^{n_{H}} = 1, if p > 2. (2.27)

It is perhaps illuminating to compute the δ-invariant in some easy non-trivial cases. Specifically, we shall prove the following result from which Dirichlet’s Theorem follows immediately:

Proposition 2.18 Let G = ℤ/pℤ × ℤ/pℤ be the elementary abelian group of order p^{2}, and put n_{{1}} = −1, n_{G} = −p and n_{H} = 1 for |H| = p. Then {n_{H}} is a character relation of G which is the only one up to integral multiples. Moreover:

a) Suppose M is a ℤG-module which is a finite cyclic abelian group. Then δ(M) := δ_{ℤ}({n_{H}}, M) = ℤ except in the case when G acts faithfully on the pprimary subgroup M^{(p)} ⊂ M. Moreover, in the exceptional case we have p = 2, 8 | m = |M| and δ(M) = 2ℤ.

b) If M is an RG-module such that G acts trivially on \overline{M} = M/M_{tor}, then

δ(M) = p^{(p−1)rk(M)} ∏_{H} χ(M/M^{H})^{2n_{H}} δ(M_{tor})^{−1}. (2.28)

In particular, δ(R) = p^{(p−1)}R.

c) Suppose M is a non-degenerate RG-module of rank 1 such that G acts nontrivially on \overline{M}. Then p = 2 and there is a unique subgroup X = ⟨σ⟩ ≤ G of order 2 such that \overline{M}^{X} = \overline{M}. Moreover,

(2.29)  δ(M) = \frac{1}{2}δ(M_{tor})χ(M/(M_{tor} + M^{X}))^{2}
= 2δ(M_{tor})χ(Ĥ^{0}(X, M_{tor}))^{2}χ(Ĥ^{0}(X, M))^{−2}.

d) Suppose M is a non-degenerate ℤG-module such that \overline{M} is as in c) and M_{tor} is as in a). In addition, assume that σ acts on M_{tor} by multiplication by −1 and that 4 | m = |M_{tor}|. Then there is a unique subgroup Y ≤ G such that M_{4}^{Y} = M_{4}, where M_{4} = {x ∈ M : 4x = 0}, and we have

δ(M_{tor})δ(M) = 2|Ĥ^{0}(Y, M)|^{−2}. (2.30)

Moreover, |Ĥ^{0}(Y, M)| = 1 if there is an element x ∈ M such that N_{Y} (x) generates M^{Y}, and |Ĥ^{0}(Y, M)| = 2 otherwise.

3. Content ideals, relative invariants and discriminants

The purpose of this section is to define and study the content ideal χ(f) attached to an R-module homomorphism f : M → N. This invariant extends the notion of a content ideal χ(T) attached to a torsion module (cf. Bourbaki[BCA], ch. VII.4.5) and reduces to the “q-invariant” q(f) of Tate[Ta] in the case that R = ℤ. We then define the relative invariant χ(f, g) of two R-module homomorphisms f, g : M → V ⊗_{K} L which is a slight refinement and generalization of the relative invariant χ(M_{1}, M_{2}) of Bourbaki[BCA], ch. VII.4.6, which is attached to two submodules M_{1}, M_{2} of a K-vector space V. This refinement will then be used to define the discriminant of an L-valued bilinear R-module (M, h).

3.1. Content ideals. As before, R is a principal ideal domain, and all R-modules are tacitly assumed to be finitely generated. If T is any torsion R-module, then its content ideal χ(T) is defined as the product

χ(T) = μ_{1} ⋯ μ_{r}R (3.1)

of its invariant factors μ_{1}|μ_{2}| ⋯ |μ_{r}. Alternatively, by the well-known formula relating invariant factors to determinants, we can also define χ(T) as the determinant χ(T) = det(f) · R, (3.2)

where

0 → R^{n} →^{f} R^{n} → T → 0, (3.3)

is a presentation of T. From the latter definition it is immediate that we have the formula χ(Coker(f)) = det(f) · R (3.4)

for every f ∈ End_{R}(R^{n}), provided we set χ(T) = 0, if T is not a torsion R-module. Furthermore, we see that the invariant χ(T) agrees with that of Bourbaki[BCA], ch. VII.4.5. In particular, it follows from [BCA] that χ(T) is additive on exact sequences, i.e. if

0 → M_{1} → M_{2} → M_{3} → 0

is an exact sequence of R-modules then we have

χ(M_{2}) = χ(M_{1}) · χ(M_{3}). (3.5)

We next want to define the content ideal χ(f) of an R-linear map f : M → N. If f is an R-isogeny in the sense that Ker(f) and Coker(f) are torsion modules, then its content ideal is the (fractional) R-ideal defined by

χ(f) = χ_{R}(f; M, N) = χ(Coker(f)) · χ(Ker(f))^{−1}. (3.6)

As above, it is useful to extend the symbol χ(f) to all f ∈ Hom_{R}(M, N) by setting χ(f) = 0 whenever f is not an R-isogeny. We thus have by (3.2) that

χ(f) = det(f)R, (3.7)

if f ∈ End_{R}(M) and M is a free R-module. More generally, if M and N are free R-modules with bases x_{1}, …, x_{m} and y_{1}, …, y_{n}, respectively, then we have

χ(f) = det(A)R, (3.8)

where A denotes the matrix of f with respect to these bases, and det(A) = 0 if the matrix A is not square.

Remark 3.1 If R′ ⊃ R is a principal ideal domain which contains R and M is an R-module, then the content ideal of the extended module M ⊗_{R} R′ is given by the formula χ_{R′}(M ⊗_{R} R′) = χ_{R}(M) · R′ (3.9)

because on the one hand M is a torsion R-module if and only if M ⊗R′ is a torsion R′-module and on the other hand the sequence (3.3) remains exact after tensoring with R′. Thus, if f : M → N is an R-linear map, then the content ideal of the extended map f ⊗ R′ : M ⊗_{R} R′ → N ⊗_{R} R′ is given by

χ_{R′}(f ⊗ R′) = χ_{R}(f) · R′. (3.10)

The following “functorial” properties of the content ideal are easily verified (cf. Tate[Ta], p. 58).

Proposition 3.2 If f : M_{1} → M_{2} and g : M_{2} → M_{3} are two R-module homomorphisms, then χ(g ∘ f) = χ(g) · χ(f). (3.11)

Corollary 3.3 Let f : M_{1} → M_{2} be an R-module homomorphism and M′_{i} ⊂ M_{i}, (i = 1, 2) two R-submodules such that f(M′_{1}) ⊂ M′_{2}. Then, if f′ = f_{|M′_{1}} : M′_{1} → M′_{2} denotes the restriction of f to M′_{1}, we have

χ(f)χ(M_{1}/M′_{1}) = χ(f′)χ(M_{2}/M′_{2}). (3.12)

Proposition 3.4 Let

\begin{aligned}0 & → & M_{1} & →^{f} & M_{2} & →^{g} & M_{3} & → & 0 \\ & & ↓ ^{h_{1}} & & ↓ ^{h_{2}} & & ↓ ^{h_{3}} \\ 0 & → & M′_{1} & →^{f′} & M′_{2} & →^{g′} & M′_{3} & → & 0 \end{aligned} (3.13)

be a commutative diagram of R-homomorphisms with exact rows. Then we have

χ(h_{2}) = χ(h_{1})χ(h_{3}), (3.14)

except possibly in the case that neither Ker(h_{3}) nor Coker(h_{1}) are torsion modules. Furthermore, if any two of the h_{i}’s are isogenies, then so is the third and (3.14) holds.

In the sequel it will often be convenient to treat the torsion and free part of a module separately. To this end we introduce the following notation.

Notation 3.5 If M is an R-module, then M_{tor} ⊂ M denotes its torsion submodule and

π_{M} : M → \overline{M} = M/M_{tor}

the quotient map onto the quotient module \overline{M} = M/M_{tor}. Furthermore, if f : M → N is a homomorphism, then we let

f_{tor} : M_{tor} → N_{tor}
\overline{f} : \overline{M} → \overline{N}

denote the induced maps.

By applying Proposition 3.4 to the exact sequences induced by π_{M} and π_{N}, we easily obtain the following formulae:

(3.15)  χ(f) = χ(f_{tor})χ(\overline{f})
(3.16)  χ(f_{tor}) = χ(M_{tor})^{−1}χ(N_{tor})
(3.17)  χ(\overline{f}) = χ(f)χ(M_{tor})χ(N_{tor})^{−1}

In particular, if M ≃ N, then we have

χ(f_{tor}) = R and χ(\overline{f}) = χ(f). (3.18)

In what follows we will need to know the relation of the content ideal χ(f) of a homomorphism f : M → N to that of its dual map f^{∗} : M^{∗} → N^{∗}, where, as before, M^{∗} = Hom_{R}(M, R).

Proposition 3.6 The content ideal χ(f^{∗}) of the dual map is related to the content ideal χ(f) of f : M → N by the formula

χ(f^{∗}) = χ(f)χ(f_{tor})^{−1} = χ(f)χ(M_{tor})χ(N_{tor})^{−1}. (3.19)

Proof. If M and N are free modules with bases x_{1}, … x_{m}, and y_{1}, … y_{n}, respectively, and A denotes the matrix of f with respect to these bases, then the matrix of f^{∗} with respect to the dual bases x_{i}^{∗} and y_{j}^{∗} is the transpose A^{t}. We thus have by (3.8) that

χ(f) = det(A)R = det(A^{t})R = χ(f^{∗}),

which is (3.19) in this case.

In the general case we note that \overline{M}^{∗} = M^{∗}, \overline{N}^{∗} = N^{∗} and \overline{f}^{∗} = f^{∗}. Applying the case just treated to \overline{f}, we therefore have

χ(f^{∗}) = χ(\overline{f^{∗}}) = χ(\overline{f}) = χ(f)χ(M_{tor})χ(N_{tor})^{−1},

the latter equality resulting from (3.17). In view of (3.16), this proves (3.19).

3.2. Relative invariants. As in the previous section, let L ⊃ K be an extension field of K = Quot(R). As was indicated there, we shall be interested in Rhomomorphisms h : M → V, where V is a (finite dimensional) L-vector space.

Definition 3.7 An R-submodule M ⊂ V is called non-degenerate with respect to L (or L-non-degenerate) if M has an R-module basis which is also an L-basis of V. Since every (finitely generated) R-submodule M ⊂ V is free (because R is a principal ideal domain), it follows that M is non-degenerate if and only if LM = V and rank_{R}(M) = dim_{L}(V). In particular, if L = K, then a nondegenerate submodule M ⊂ V is just a lattice in the sense of Bourbaki[BCA], VII.4.1.

If M is an R-module and f : M → V is R-linear, then f is said to be nondegenerate (with respect to L) if f(M) is a non-degenerate submodule of V and Ker(f) is a torsion module (and hence Ker(f) = M_{tor}).

Definition 3.8 If M_{1} and M_{2} are two non-degenerate submodules of V, then their relative invariant is defined by

χ(M_{1}, M_{2}) = χ_{R,L}(M_{1}, M_{2}; V) = det(α)R, (3.20)

where α ∈ Aut_{L}(V) is any automorphism of V such that

α(M_{2}) = M_{1}. (3.21)

Furthermore, if f_{i} : M_{i} → V, i = 1, 2 are two R-linear maps which are nondegenerate, then their relative invariant is defined by

χ(f_{1}, f_{2}) = χ(f_{1}, f_{2}; V) = χ((M_{1})_{tor})^{−1}χ((M_{2})_{tor})χ(f_{1}(M_{1}), f_{2}(M_{2})). (3.22)

Remark 3.9 a) Note that in the above definition the right hand side of (3.20) does not depend on the choice of α, for if β is another choice satisfying (3.21), then β = γ_{1} ∘ α ∘ γ_{2} with γ_{i} ∈ Aut_{R}(M_{i}), and so det_{L}(β)R = det_{R}(γ_{1}) det_{L}(α) det_{R}(γ_{2})R = det(α)R since det(γ_{i}) ∈ R^{×}. We thus see by [BCA], VII.4.6, equation (6), that in the case that L = K the relative invariant χ(M_{1}, M_{2}) coincides with that of Bourbaki.

b) The above Definition 3.8 may be extended to the case that M_{1} is degenerate by observing that since M_{2} is non-degenerate, we can always find α ∈ End_{L}(V) such that (3.21) holds, at least if rank_{R}(M_{1}) ≤ dim(V) (see c) below). Thus Definition 3.20 still makes sense (provided we set we set χ(M_{1}, M_{2}) = 0 when rank(M_{1}) > dim(V)), and we obtain

(3.23) χ(M_{1}, M_{2}) = 0 ⟺ M_{1} is degenerate.

In a similar manner we can extend the symbol χ(f_{1}, f_{2}) to the case that f_{2} is non-degenerate and f_{1} is arbitrary by setting

χ(f_{1}, f_{2}) = 0 (3.24) whenever f_{1} is degenerate.

c) More explicitly, Definition 3.8 may be written as follows. Let x_{1}, …, x_{m} and y_{1}, …, y_{n} be R-bases of the submodules M_{1} and M_{2}, respectively. Since M_{2} is L-non-degenerate by hypothesis, we can find c_{ij} ∈ L such that

x_{i} = ∑_{j=1}^{n} c_{ij}y_{j}, for 1 ≤ i ≤ m, (3.25)

and so we have χ(M_{1}, M_{2}) = det(c_{ij})R, (3.26)

where we set det(C) = 0 if the matrix C is not square.

The following properties of the symbol χ(f_{1}, f_{2}) follow immediately from the definitions:

(3.27)  χ(f_{2}, f_{1}) = χ(f_{1}, f_{2})^{−1};
(3.28)  χ(f_{1}, f_{3}) = χ(f_{1}, f_{2})χ(f_{2}, f_{3});
(3.29)  χ(α ∘ f_{1}, f_{2})) = det(α)χ(f_{1}, f_{2});

here, as before, the f_{i} are non-degenerate R-linear maps to V and α ∈ Aut_{L}(V) is an automorphism. There are analogous properties for the symbol χ(M_{1}, M_{2}), but we do not need these here.

The relative invariants are closely connected to the content ideals defined above. To see this, suppose first that M_{1} and M_{2} are two non-degenerate submodules of V with M_{1} ⊂ M_{2}. Since both have the same rank, the quotient module M_{2}/M_{1} is a torsion R-module and we have by comparing (3.4) with (3.20):

χ(M_{1}, M_{2}) = χ(M_{2}/M_{1}). (3.30)

More generally, if M_{1} and M_{2} are two non-degenerate submodules of V which are commensurable in the sense that there exists a non-degenerate submodule M ⊃ M_{1} + M_{2} containing M_{1} and M_{2}, then we have by (3.30) and (3.28):

χ(M_{1}, M_{2}) = χ(M/M_{1}) · χ(M/M_{2})^{−1}. (3.31)

Note that in this case M_{3} = M_{1} ∩ M_{2} (as well as M_{1} + M_{2}) is also non-degenerate, and so we also have

χ(M_{1}, M_{2}) = χ(M_{1}/M_{3})^{−1} · χ(M_{2}/M_{3}). (3.32)

From these formulae we see that if f_{i} : M_{i} → V, i = 1, 2 are two non-degenerate R-linear maps which are commensurable in the sense that M = f_{1}(M_{1}) + f_{2}(M_{2}) is a non-degenerate submodule of V, then we have χ(f_{1}, f_{2}) = χ(f_{1}; M_{1}, M) · χ(f_{2}; M_{2}, M)^{−1}, (3.33)

where we regard the f_{i} as R-linear maps f_{i} : M_{i} → M ⊂ V. In view of this formula, which follows immediately from the definition (3.22) and from (3.31), many of the properties of content ideals can be transferred to the relative invariants. For example, if g_{i} : N_{i} → M_{i} are R-isogenies, then by (3.11) and (3.33) we obtain the formula

χ(f_{1} ∘ g_{1}, f_{2} ∘ g_{2}) = χ(g_{1})χ(g_{2})^{−1}χ(f_{1}, f_{2}), (3.34)

at least when f_{1} and f_{2} are commensurable. However, this formula is easily verified to be correct in the general case. Indeed, since f_{i} and f_{i} ∘ g_{i} are clearly commensurable, we obtain by (3.28) and the proven case of (3.34) that

χ(f_{1} ∘ g_{1}, f_{2} ∘ g_{2}) = χ(f_{1} ∘ g_{1}, f_{1})χ(f_{1}, f_{2})χ(f_{2}, f_{2} ∘ g_{2})
= χ(g_{1})χ(f_{1}, f_{1})χ(f_{1}, f_{2})χ(g_{2})^{−1}χ(f_{2}, f_{2})
= χ(g_{1})χ(g_{2})^{−1}χ(f_{1}, f_{2}),

which proves (3.34) in general. Note that this formula continues to be true if we drop the condition that f_{1} be an isogeny.

3.3. Discriminants of bilinear R-modules. Recall from the previous section that an L-valued, bilinear R-module is a pair (M, h) consisting of an R-module M and an R-linear map

h : M → M^{∗} ⊗ L = Hom_{R}(M, L).

We can also describe h by its associated bilinear map

β_{h} : M × M → L

which is given by the formula

β_{h}(x, y) = h(x)(y), (3.35)

for β_{h} and h determine each other via the formula (3.35). (Usually h = β̂ is called the adjoint map of β = β_{h}.) If no confusion results, then we will just write

h(x, y) = β_{h}(x, y).

Definition 3.10 The discriminant of an L-valued bilinear R-module (M, h) is defined as the relative invariant

disc_{R,L}(M, h) = χ(h, j_{M,L}; V) (3.36)

where

j_{M,L} = id_{M^{∗}} ⊗ L : M^{∗} → V = M^{∗} ⊗ L

denotes the canonical map j_{M,L}(m) = m ⊗ 1. Note that j is trivially nondegenerate (in the sense of Definition 3.7), so this relative invariant is always defined.

Remark 3.11 a) If M = \overline{M} is a free R-module with basis x_{1}, …, x_{n}, then its discriminant is given by the determinant of the associated Gram matrix, i.e. we have the formula disc(M, h) = det(h(x_{i}, x_{j})) · R, (3.37)

which is the usual definition of the discriminant. To see this, note that if we let x_{1}^{∗}, …, x_{n}^{∗} ∈ M^{∗} denote the dual basis of x_{1}, …, x_{n}, then x_{1}^{∗} ⊗ 1, …, x_{n}^{∗} ⊗ 1 is an R-module basis of M_{2} = j(M^{∗}) ⊂ V = M^{∗} ⊗ L and we have

h(x_{i}) = ∑_{j=1}^{n} h(x_{i}, x_{j})x_{j}^{∗} ⊗ 1.

Thus, if h(x_{1}), …, h(x_{n}) is a basis of M_{1} = h(M), then (3.37) follows from (3.26). On the other hand, if h(x_{1}), …, h(x_{n}) are linearly dependent, then we have

det(h(x_{i}, x_{j})) = 0 = disc(M, h),

where the latter equality holds by definition since M_{1} is degenerate, and so (3.37) holds in all cases.

b) For an arbitrary bilinear R-module (M, h) we have

disc(M, h) = χ(M_{tor})^{−1}disc(\overline{M}, \overline{h}), (3.38)

where \overline{h} : \overline{M} → \overline{M}^{∗} ⊗ L = M^{∗} ⊗ L denotes the induced map (cf. Notation 3.5). This is true because we have h = \overline{h} ∘ π_{M} and j_{M,L} = j_{\overline{M},L} and so we obtain from (3.34) that disc(M, h) = χ(h, j_{M,L}) = χ(π_{M})χ(\overline{h}, j_{\overline{M},L}) = χ(M_{tor})^{−1}disc(\overline{M}, \overline{h}).

c) It is natural to call (M, h) non-degenerate if h : M → M^{∗} ⊗ L has this property. We then have by Remark 3.9b):

(M, h) is non-degenerate ⟺ disc(M, h) ≠ 0.

For future reference, let us note that discriminants are compatible with basechange:

Proposition 3.12 Let L′ ⊃ L be an extension field of L and let R′ be a principal ideal domain with R ⊂ R′ ⊂ L′. Then for any bilinear R-module (M, h) we have

disc_{R′,L′}(M ⊗_{R} R′, h ⊗ R′) = disc_{R,L}(M, h) · R′. (3.39)

Proof. Write M′ = M ⊗ R′. By flatness we have that M′_{tor} = M_{tor} ⊗ R′ , and so we obtain from Remark 3.1 that

χ(M′_{tor}) = χ(M) · R′.

Furthermore, since \overline{M}′ = \overline{M} ⊗ R′, it follows from (3.37) that

disc_{R′L′}(\overline{M′}, \overline{h} ⊗ R′) = disc_{R,L}(\overline{M}, \overline{h}) · R′,

and so the assertion follows in view of (3.38).

Corollary 3.13 For any bilinear R-module (M, h) we have

disc_{R,L}(M, h) = ⋂_{𝔭} disc_{R_{𝔭},L}(M_{𝔭}, h_{𝔭}), (3.40)

where the intersection is over all non-zero prime ideals 𝔭 of R.

Proof. We have

disc_{R,L}(M, h) = ⋂ disc_{R,L}(M, h) · R_{𝔭} = ⋂ disc_{R_{𝔭},L}(M_{𝔭}, h_{𝔭}),

where the latter follows from (3.39).

There are various operations that can be performed on bilinear R-modules (M, h). These will now be considered in turn and their effect on discriminants determined.

The first of these is twisting an L-valued bilinear module (M, h) by an automorphism α ∈ Aut_{L}(M^{∗} ⊗ L):

α · (M, h) = (M, αh) = (M, α ∘ h). (3.41)

A special case of this is the case that α = c · id is multiplication by an element c ∈ L; in this case we shall also write

(M, h)(c) = (c · id)(M, h).

By formula (3.29) we have:

disc(M, αh) = det(α)disc(M, h); (3.42)

in particular,

disc((M, h)(c)) = disc((M, ch)) = c^{rk(M)}disc(M, h), if c ∈ L. (3.43)

Another operation is the pullback of (M, h) with respect to two R-linear maps f, g ∈ Hom_{R}(N, M); it is defined by

(f, g)^{∗}(M, h) = (N, (g^{∗} ⊗ L) ∘ h ∘ f). (3.44)

Thus, the associated bilinear maps are related by the formula

β_{h′} = β_{h} ∘ (f × g), (3.45)

where h′ = (g^{∗} ⊗ L) ∘ h ∘ f. If f = g then we shall write

f^{#}(M, h) = (N, f^{#}h) = (f, f)^{∗}(M, h). (3.46)

Proposition 3.14 If dim(M ⊗ K) = dim(N ⊗ K), then we have

(3.47) disc((f, g)^{∗}(M, h)) = χ(f)χ(g^{∗})disc(M, h)
= χ(f)χ(g)χ(N_{tor})χ(M_{tor})^{−1}disc(M, h).

Proof. Since the second formula follows directly from Proposition 3.6, it is enough to verify the first. Here we have by definition and (3.34)

disc((f, g)^{∗}(M, h)) = χ((g^{∗} ⊗ L) ∘ h ∘ f, j_{N,L}) = χ(f)χ((g^{∗} ⊗ L) ∘ h, j_{N,L}).

Now if g is not an isogeny, then neither is g^{∗}, and hence (g^{∗} ⊗ L) ∘ h is degenerate because dim(M^{∗} ⊗ L) = dim(N^{∗} ⊗ L). Thus, both sides of (3.47) are zero in this case, and so we may assume henceforth that g is an isogeny. Then (g^{∗}⊗L)∘j_{M,L} = j_{N,L} ∘ g^{∗} : M^{∗} → N^{∗} ⊗ L is non-degenerate, so we have by (3.28) and (3.34)

χ((g^{∗} ⊗ L) ∘ h, j_{N,L}) = χ((g^{∗} ⊗ L) ∘ h, (g^{∗} ⊗ L) ∘ j_{M,L})χ(j_{N,L} ∘ g^{∗}, j_{N,L})
= χ(h, j_{M,L})χ(g^{∗}),

and hence the formula follows.

Corollary 3.15 If (M, h) is a bilinear R-module and M′ ⊂ M is a submodule of finite index (i.e. M′ ⊗ K = M ⊗ K), then

disc(M′, h_{|M′}) = χ(M/M′)^{2}χ(M_{tor})^{−1}χ(M′_{tor})disc(M, h). (3.48)

Proof. This follows from (3.47) by taking f = g : M′ → M as the inclusion map; recall that then χ(f) = χ(M/M′) by definition.

The last operations which we consider here are the direct sum and tensor product of two bilinear R-modules (M_{1}, h_{1}) and (M_{2}, h_{2}). These are defined by the formulae

(3.49)  (M_{1}, h_{1}) ⊕ (M_{2}, h_{2}) = (M_{1} ⊕ M_{2}, h_{1} ⊕ h_{2})
(3.50)  (M_{1}, h_{1}) ⊗ (M_{2}, h_{2}) = (M_{1} ⊗ M_{2}, h_{1} ⊗ h_{2}),

where the maps h_{1} ⊕ h_{2} and h_{1} ⊗ h_{2} are the compositions

M_{1} ⊕ M_{2} →^{h_{1}⊕h_{2}} (M_{1}^{∗} ⊗ L) ⊕ (M_{2}^{∗} ⊗ L) →^{ρ} (M_{1} ⊕ M_{2})^{∗} ⊗ L, (3.51)

M_{1} ⊗ M_{2} →^{h_{1}⊗h_{2}} (M_{1}^{∗} ⊗ L) ⊗ (M_{2}^{∗} ⊗ L) →^{μ} (M_{1} ⊗ M_{2})^{∗} ⊗ L, (3.52)

where ρ : M_{1}^{∗} ⊕ M_{2}^{∗} → (M_{1} ⊕ M_{2})^{∗} and μ : M_{1}^{∗} ⊗ M_{2}^{∗} → (M_{1} ⊗ M_{2})^{∗} are the canonical maps (cf. Bourbaki[BA], II.2.6 and II.4.4). Thus, for their associated bilinear forms we have:

(3.53)  β_{h_{1}⊕h_{2}}(x_{1} ⊕ x_{2}, y_{1} ⊕ y_{2}) = β_{h_{1}}(x_{1}, y_{1}) + β_{h_{2}}(x_{2}, y_{2})
(3.54)  β_{h_{1}⊗h_{2}}(x_{1} ⊗ x_{2}, y_{1} ⊗ y_{2}) = β_{h_{1}}(x_{1}, y_{1}) · β_{h_{2}}(x_{2}, y_{2}).

Proposition 3.16 We have

disc((M_{1}, h_{1}) ⊕ (M_{2}, h_{2})) = disc(M_{1}, h_{1}) · disc(M_{2}, h_{2}), (3.55) disc((M_{1}, h_{1}) ⊗ (M_{2}, h_{2})) = (3.56)

χ((M_{1})_{tor} ⊗ (M_{2})_{tor})^{−1}disc(M_{1}, h_{1})^{d_{2}} · disc(M_{2}, h_{2})^{d_{1}},

where d_{i} = dim_{K}(M_{i} ⊗ K).

Proof. If M_{1} and M_{2} are free R-modules, then the discriminants are the determinants of the corresponding Gram matrices and hence these formulae follow from the usual identities for determinant of a direct sum (respectively, of a tensor product) of matrices.

Now let M_{1} and M_{2} be arbitrary. Then (M_{1} ⊕ M_{2})_{tor} = (M_{1})_{tor} ⊕ (M_{2})_{tor} so χ((M_{1} ⊕ M_{2})_{tor}) = χ((M_{1})_{tor}) · χ((M_{2})_{tor}) and hence the first equation follows.

Next, since M_{i} ≃ (M_{i})_{tor} ⊕ \overline{M}_{i} we have

(M_{1} ⊗ M_{2})_{tor} = ((M_{1})_{tor} ⊗ (M_{2})_{tor}) ⊕ (\overline{M}_{1} ⊗ (M_{2})_{tor}) ⊕ ((M_{1})_{tor} ⊗ \overline{M}_{2}) ≃ ((M_{1})_{tor} ⊗ (M_{2})_{tor}) ⊕ ((M_{1})_{tor})^{rkM_{2}} ⊕ ((M_{2})_{tor})^{rkM_{1}},

and so

disc(M_{1} ⊗ M_{2}) = χ((M_{1} ⊗ M_{2})_{tor})^{−1}disc(\overline{M}_{1} ⊗ \overline{M}_{2}) = χ((M_{1})_{tor} ⊗(M_{2})_{tor})^{−1}·χ((M_{1})_{tor}))^{−d_{2}} ·χ((M_{2})_{tor})^{−d_{1}} ·disc(\overline{M}_{1})^{d_{2}} ·disc(\overline{M}_{2})^{d_{1}} = χ((M_{1})_{tor} ⊗ (M_{2})_{tor})^{−1}disc(M_{1})^{d_{2}}disc(M_{2})^{d_{1}},

which proves the second equation.

Corollary 3.17 If (M_{1}, h_{1}) and (M_{2}, h_{2}) are non-degenerate bilinear R-modules, then so are (M_{1}, h_{1}) ⊕ (M_{2}, h_{2}) and (M_{1}, h_{1}) ⊗ (M_{2}, h_{2}).

Proof. In view of Remark 3.11c), the hypothesis means that disc(M_{i}, h_{i}) ≠ 0, for i = 1, 2. By formula (3.55), respectively (3.56), we see that disc((M_{1}, h_{1}) ⊕ (M_{2}, h_{2})) ≠ 0, respectively disc((M_{1}, h_{1}) ⊗ (M_{2}, h_{2})) ≠ 0, and so (M_{1}, h_{1}) ⊕ (M_{2}, h_{2}) and (M_{1}, h_{1}) ⊗ (M_{2}, h_{2}) are non-degenerate by Remark 3.11c) again.

4. Induced hermitian R[G]-modules

In this section we shall review the definition of a hermitian RG-module and study the processes of (co)induction and restriction. In particular, we shall formulate and prove several versions of a (generalized) Frobenius reciprocity theorem for hermitian RG-modules.

4.1. Hermitian RG-modules. Recall from section 2 that an L-valued, hermitian RG-module is a bilinear R-module (M, h) such that M is a left RG-module and h : M → M^{∗} ⊗ L is RG-linear. In terms of the associated bilinear map β_{h}, the RG-linearity is equivalent to the G-equivariance condition

β_{h}(gm, gm′) = β_{h}(m, m′), ∀g ∈ G, m, m′ ∈ M. (4.1)

The above use of the adjective “hermitian”, which follows in part that of Scharlau[Sch], p. 244, is justified by the fact that h is essentially a sesquilinear RG-module homomorphism with respect to the canonical involution ^{∗} on RG given by

(∑_{g∈G} a_{g}g)^{∗} = ∑_{g∈G} a_{g}g^{−1}.

More generally, if ^{∗} is any involution on R which extends to one on L, then it can be extended to one on RG and we can define hermitian RG-modules relative to this involution. It is easily checked that all the results below readily extend to this more general situation. However, since all the applications only use the case ∗ = id_{R} (and since the general case is more awkward to state) we restrict attention to this situation.

The following example will play a major role in the applications to character relations (cf. section 7).

Example 4.1 Permutation modules. Let S be a finite set on which G acts from the left, and let

R[S] = ⨁_{s∈S} Rs

denote the associated permutation module. Then M = R[S] comes equipped with a natural hermitian RG-module structure h_{S} : M → M^{∗} defined by

h_{S}(s) = s^{∗}, for all s ∈ S, (4.2)

where {s^{∗}} denotes the dual basis of M^{∗} with respect to {s}_{s∈S}. It is clear that (R[S], h_{S}) is a unimodular hermitian RG-module in the sense that h_{S} : R[S] → R[S]^{∗} is an RG-isomorphism. Note that if we take S = {s}, a singleton, then we obtain the trivial hermitian RG-module (R, id) = (R[S], h_{S}) as a special case.

It should also be remarked that the hermitian RG-module (R[S], h_{S}) is uniquely determined by the G-set S, for each G-set isomorphism S ≃ S′ extends uniquely to an RG-isometry (R[S], h_{S}) ≃ (R[S′], h_{S′}).

It is immediate that the operations on bilinear R-modules which were introduced in the previous section readily extend to RG-modules. For the twisting, direct sum and tensor product operations no further assumptions are necessary but for the pullback we need to assume that the homomorphisms f, g ∈ Hom_{R}(N, M) are actually RG-linear so that (f, g)^{∗}(M, h) is again a hermitian RG-module.

4.2. Restriction and Induction. We now turn to operations on hermitian RG-modules which result from changing the group.

Definition 4.2 Let f : G_{1} → G_{2} be a group homomorphism.

a) If (M_{2}, h_{2}) is a hermitian RG_{2}-module, then its restriction is defined as

Res_{f}(M_{2}, h_{2}) = (M_{2}, h_{2}),

where we view M_{2} as an RG_{1}-module via f. Then h_{2} is automatically RG_{1}-linear, so Res_{f}(M_{2}, h_{2}) is a hermitian RG_{1}-module.

b) If (M_{1}, h_{1}) is a hermitian RG_{1}-module, then the induced hermitian RG_{2}- module

Ind_{f}(M_{1}, h_{1}) = (Ind_{f}(M_{1}), ind_{f}(h_{1}))

is defined as follows. The underlying RG_{2}-module is

Ind_{f}(M_{1}) = RG_{2} ⊗_{RG_{1}} M_{1},

which is endowed with the hermitian structure ind_{f}(h_{1}) defined by the formula

ind_{f}(h_{1})(g_{2} ⊗ m, g′_{2} ⊗ m′) = h_{1}(ρ_{f}(g_{2}, g′_{2})m, m′), (4.3)

where g_{2}, g′_{2} ∈ G_{2}, m, m′ ∈ M_{1} and where ρ_{f} : RG_{2}×RG_{2} → RG_{1} is the R-bilinear map defined by

ρ_{f}(g_{2}, g′_{2}) = ∑_{g_{1}∈f^{−1}(g_{2}^{′−1}g_{2})} g_{1}, for g_{2}, g′_{2} ∈ G_{2}. (4.4)

If the homomorphism f : G_{1} → G_{2} is clear from the context, then we shall also write Res_{G_{1}}^{G_{2}}(M_{2}, h_{2}) = Res_{f}(M_{2}, h_{2}) and Ind_{G_{1}}^{G_{2}}(M_{1}, h_{1}) = Ind_{f}(M_{1}, h_{1}).

Remark 4.3 a) The fact that the above formula (4.3) actually defines a (unique) hermitian structure h_{2} = ind_{f}(h_{1}) on M_{2} = Ind_{f}(M_{1}) requires some justification. To this end, we first observe that the function ρ_{f} satisfies the following properties:

(4.5)  ρ_{f}(g_{2}f(g_{1}), g′_{2}) = ρ_{f}(g_{2}, g′_{2})g_{1},
(4.6)  ρ_{f}(g_{2}, g′_{2}f(g_{1})) = g_{1}^{−1}ρ_{f}(g_{2}, g′_{2}),
(4.7)  ρ_{f}(gg_{2}, gg′_{2}) = ρ_{f}(g_{2}, g′_{2}),

where g_{1} ∈ G_{1} and g, g_{2}, g′_{2} ∈ G_{2}.

From the first two properties and the fact that h_{1} is RG_{1}-linear it follows that the function

h̃_{2}(g_{2}, m, g′_{2}, m′) = h_{1}(ρ_{f}(g_{2}, g′_{2})m, m′)

on RG_{2} × M × RG_{2} × M satisfies

(4.8)  h̃_{2}(g_{2}f(g_{1}), m, g′_{2}, m′) = h̃_{2}(g_{2}, g_{1}m, g′_{2}, m′),
(4.9)  h̃_{2}(g_{2}, m, g′_{2}f(g_{1}), m′) = h̃_{2}(g_{2}, m, g′_{2}, g_{1}m′),

and hence induces a unique function h_{2} on (RG_{2} ⊗_{RG_{1}} M)×(RG_{2} ⊗_{RG_{1}} M) which satisfies (4.3). Furthermore, the third property shows that h_{2} is G_{2}-equivariant.

b) The above formula (4.3) defines the hermitian structure on Ind_{f}(M) in terms of its associated bilinear form. However, it is not difficult to write down the hermitian structure h_{2} = ind_{f}(h_{1}) : M_{2} = RG_{2} ⊗_{RG_{1}} M_{1} → M_{2}^{∗} ⊗ L directly; in fact, we clearly have

ind_{f}(h_{1}) = ρ_{f,M_{1}} ⊗ L ∘ id_{RG_{2}} ⊗ h_{1}, (4.10)

where the map

ρ_{f,M_{1}} : RG_{2} ⊗ M_{1}^{∗} → (RG_{2} ⊗ M)^{∗}

is defined by the rule

ρ_{f,M_{1}}(g_{2} ⊗ m^{∗})(g′_{2} ⊗ m′) = ρ_{f}(g_{2}, g′_{2})m^{∗}(m′) = ∑_{g_{1}∈f^{−1}(g_{2}^{′−1}g_{2})} m^{∗}(g_{1}^{−1}m′); (4.11)

here, g_{2}, g′_{2} ∈ G_{2}, m′ ∈ M and m^{∗} ∈ M^{∗}. Note that properties (4.5) and (4.6) guarantee that ρ_{f,M_{1}} is well-defined and (4.7) shows that it is RG_{2}-linear.

For later reference, let us also observe that the rule

ϕ(g′ ⊗ m)(g) = ρ_{f}(g′, g^{−1})m, (4.12)

where g, g′ ∈ G_{2}, m ∈ M, defines by properties (4.5) – (4.7) an RG_{2}-linear map

ϕ = ϕ_{f,M} : RG_{2} ⊗_{RG_{1}} M → Hom_{RG_{1}}(RG_{2}, M).

Example 4.4 a) Let H ≤ G be a subgroup, and let f : H → G denote the inclusion map. If (M, h) is a hermitian RH-module, then its induced hermitian RG-module is Ind_{H}^{G}(M, h) = (Ind_{H}^{G}(M), ind_{H}^{G}(h)), where Ind_{H}^{G}(M) = RG⊗_{RH} M denotes the usual induced module and h′ = ind_{H}^{G}(h) is given by the formula

h′(∑g ⊗ m_{g}, ∑g ⊗ m′_{g}) = ∑ h(m_{g}, m′_{g}), (4.13)

where all sums are over a system T = {g_{t}}_{t∈T} ⊂ G of left coset representatives of G/H. This follows immediately from (4.3) since we have

ρ_{f}(g_{t}, g_{s}) = δ(g_{t}, g_{s})

for g_{t}, g_{s} ∈ T . Note that in this case the map

ρ_{f,M} : Ind(M^{∗}) → (Ind(M))^{∗}

is the same as that defined by [CR1], Proposition (10.28), and is therefor an isomorphism if M is torsionfree.

b) Let S = G/H denote the left coset space of G with respect to H ≤ G. Then with the notation of Example 4.1 we have by (4.13) that

(R[S], h_{S}) = Ind_{H}^{G}(R, id). (4.14)

More generally, for any G-set S we have

(R[S], h_{S}) = ⨁_{s∈G\S} Ind_{G_{s}}^{G}(R, id), (4.15)

where G_{s} = {g ∈ G : gs = s} denotes the stabilizer of s ∈ S.

c) Let f : G_{1} = G → G_{2} = {1} denote the trivial map, and let (M, h) be a hermitian RG-module. Then

Ind_{f}(M) = R ⊗_{RG} M = H_{0}(G, M) = M_{G} = M/D_{G}M (4.16)

is the 0-th homology module of M, where D_{G}M is the submodule of M generated by the elements gm − m, g ∈ G; cf. Serre[Se], ch. VII.4. The hermitian structure on H_{0}(G, M) = M/D_{G}M is given by

h_{G}(m, m′) = h(N_{G}m, m′), for m, m′ ∈ M, (4.17)

where N_{G} = ∑_{g∈G} g = ρ_{f}(1).

In the sequel we shall denote this example by

H_{0}(G, (M, h)) = (M_{G}, h_{G}) = Ind_{f}(M, h). (4.18)

Proposition 4.5 Let f : G_{1} → G_{2} be an injective group homomorphism, and let (M_{i}, h_{i}) be a hermitian RG_{i}-module for i = 1, 2. Then there is a natural isometry

Ind_{f}((M_{1}, h_{1}) ⊗_{R} Res_{f}(M_{2}, h_{2})) ≃ (Ind_{f}(M_{1}, h_{1})) ⊗_{R} (M_{2}, h_{2}). (4.19)

Proof. Fix a system T of left coset representatives of G/Im(f), and consider the R-linear map

φ = φ_{T} : RG_{2} ⊗_{RG_{1}} (M_{1} ⊗_{R} M_{2}) → (RG_{2} ⊗_{RG_{1}} M_{1}) ⊗_{R} M_{2}

defined by φ(t ⊗ (m_{1} ⊗ m_{2})) = (t ⊗ m_{1}) ⊗ tm_{2}, where t ∈ T , m_{i} ∈ M_{i}, i = 1, 2; such a map exists and is in fact an isomorphism of R-modules because

RG_{2} ⊗_{RG_{1}} (M_{1} ⊗_{R} M_{2}) = ⨁_{t∈T} t ⊗ (M_{1} ⊗_{R} M_{2}),
(RG_{2} ⊗_{RG_{1}} M_{1}) ⊗_{R} M_{2} = ⨁_{t∈T}(t ⊗ M_{1}) ⊗_{R} M_{2}.

By a calculation as in [Hu], p. 60, one checks that φ is RG_{2}-linear, and hence is an isomorphism of RG_{2}-modules. To check that it is also an isometry, let t, t′ ∈ T , m_{i}, m′_{i} ∈ M_{i}. Then, using the fact that ρ_{f}(t, t′) = δ(t, t′) = 0 or 1, we obtain:

(ind_{f}(h_{1}) ⊗ h_{2})(φ(t ⊗ (m_{1} ⊗ m_{2})), φ(t′ ⊗ (m′_{1} ⊗ m′_{2})))
= (ind_{f}(h_{1}) ⊗ h_{2})((t ⊗ m_{1}) ⊗ tm_{2}, (t′ ⊗ m′_{1}) ⊗ t′m′_{2})
= ind_{f}(t ⊗ m_{1}, t′ ⊗ m′_{1}) · h_{2}(tm_{2}, t′m′_{2})
= h_{1}(ρ_{f}(g, g′)m_{1}, m′_{1}) · h_{2}(m_{2}, m′_{2})
= (h_{1} ⊗ h_{2})(ρ_{f}(t, t′)(m_{1} ⊗ m_{2}), m′_{1} ⊗ m′_{2})
= ind_{f}(h_{1} ⊗ h_{2})(t ⊗ (m_{1} ⊗ m_{2}), t′ ⊗ (m′_{1} ⊗ m′_{2})),

which proves that φ is an isometry.

The process of induction is functorial in the following sense.

Proposition 4.6 Let f_{1} : G_{1} → G_{2} and f_{2} : G_{2} → G_{3} be two group homomorphisms, and let (M_{1}, h_{1}) be a hermitian RG_{1}-module. Then there is a natural isometry

Ind_{f_{2}∘f_{1}}(M_{1}, h_{1}) ≃ Ind_{f_{2}}(Ind_{f_{1}}(M_{1}, h_{1})).

Proof. Consider the map

ψ : RG_{3} ⊗_{RG_{2}} (RG_{2} ⊗_{RG_{1}} M) → RG_{3} ⊗_{RG_{1}} M

defined by ψ(g_{3} ⊗ (g_{2} ⊗ m)) = g_{3}f_{2}(g_{2}) ⊗ m, which is clearly an isomorphism of RG_{3}-modules. To see that it is an isometry, we first note that the following identity holds:

ρ_{f_{2}∘f_{1}}(g_{3}f_{2}(g_{2}), g′_{3}f_{2}(g′_{2})) = ρ_{f_{1}}(ρ_{f_{2}}(g_{3}, g′_{3})g_{2}, g′_{2}). (4.20)

Using this, we obtain

ind_{f_{2}∘f_{1}}(h_{1})(ψ(g_{3} ⊗ (g_{2} ⊗ m)), ψ(g′_{3} ⊗ (g′_{2} ⊗ m′)))
= ind_{f_{2}∘f_{1}}(h_{1})(g_{3}f_{2}(g_{2}) ⊗ m, g′_{3}f_{2}(g′_{2}) ⊗ m′)
= h_{1}(ρ(g_{3}f_{2}(g_{2}), g′_{3}f_{2}(g′_{2}))m, m′)
= h_{1}(ρ_{f_{1}}(ρ_{f_{2}}(g_{3}, g′_{3})g_{2}, g′_{2})m, m′)
= ind_{f_{1}}(h_{1})(ρ_{f_{1}}(g_{3}, g′_{3})(g_{2} ⊗ m), g′_{2} ⊗ m′)
= ind_{f_{2}}(ind_{f_{1}}(h_{1}))(g_{3} ⊗ (g_{2} ⊗ m), g′_{3} ⊗ (g′_{2} ⊗ m′)),

which shows that ψ is an isometry.

Corollary 4.7 Let f : G_{1} → G_{2} be an injective group homomorphism and let (M_{i}, h_{i}) be a hermitian RG_{i}-module. Then there is an isometry

H_{0}(G_{2}, Ind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2})) ≃ H_{0}(G_{1}, (M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})). (4.21)

Proof. Let f′ : G_{2} → G_{3} = {1} denote the trivial map of G_{2}. Then f′∘f : G_{1} → G_{3} is the trivial map of G_{1}, so we have by Propositions 4.5 and 4.6 (and Example 4.4c))

H_{0}(G_{1}, (M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})) = Ind_{f′∘f}((M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2}))
≃ Ind_{f′}(Ind_{f}((M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2}))
≃ Ind_{f′}(Ind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2}))
= H_{0}(G_{2}, (Ind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2}))).

The above Corollary 4.7 may be viewed as a (primitive) version of Frobenius Reciprocity, for over a field R = K one can easily deduce the usual reciprocity theorem by taking dimensions of the underlying vector spaces. For our purposes, however, this version is not so useful because the applications require the module

H^{0}(G, M) = M^{G} = inv_{G}(M) (4.22)

of G-invariants of M in place of H_{0}(G, M). Over a field R = K these two modules are the same, but in general they are only connected by the exact sequence

0 → Ĥ^{−1}(G, M) → H_{0}(G, M) →^{N̄_{G}} H^{0}(G, M) → Ĥ^{0}(G, M) → 0, (4.23)

where N̄_{G} is the map induced by the endomorphism m → N_{G}m of M, and the Ĥ^{q} denote the Tate cohomology groups.

Notation 4.8 If (M, h) is a hermitian RG-module, then, as in section 2, we shall endow the module H^{0}(G, M) = Inv_{G}(M) with the hermitian structure

inv_{G}(h)(m, m′) = \frac{1}{|G|}h(m, m′). (4.24)

The resulting bilinear R-module is denoted by

H^{0}(G, (M, h)) = Inv_{G}(M, h) = (Inv_{G}(M), inv_{G}(h)).

Remark 4.9 The normalization in (4.24) has been chosen so that the hermitian structure on H^{0}(G, M) pulls back to that on H_{0}(G, M) via N̄_{G}:

N̄_{G}^{#}(H^{0}(G, (M, h))) = H_{0}(G, (M, h)). (4.25)

This follows immediately from formula (4.17) because by the G-equivariance of h we have

inv_{G}(h)(N_{G}m, N_{G}m′) = \frac{1}{|G|} ∑_{g∈G} h(N_{G}m, gm′) = h(N_{G}m, m′).

4.3. Coinduction. We had seen above that the bilinear module H_{0}(G, (M, h)) is a special case of the general construction of induced modules. In a similar manner, the “dual” module H^{0}(G, (M, h)) is a special case of the dual construction of coinduced modules which we introduce now.

Definition 4.10 Let f : G_{1} → G_{2} be a homomorphism of groups, and let (M_{1}, h_{1}) be a hermitian RG_{1}-module. Then the coinduced hermitian RG_{2}-module

Coind_{f}(M_{1}, h_{1}) = (Coind_{f}(M_{1}), coind_{f}(h_{1}))

is defined as follows. The underlying RG_{2}-module is

Coind_{f}(M_{1}) = Hom_{RG_{1}}(RG_{2}, M_{1}),

which is endowed with the hermitian structure coind_{f}(h_{1}) defined by the formula

coind_{f}(h_{1})(f_{1}, f_{2}) = \frac{1}{|G_{1}|} ∑_{g_{2}∈G_{2}} h_{1}(f_{1}(g_{2}), f_{2}(g_{2})), (4.26)

where f_{1}, f_{2} ∈ Hom_{RG_{1}}(RG_{2}, M_{1}).

Remark 4.11 a) The above bilinear form coind_{f}(h_{1}) is G_{2}-equivariant because

coind_{f}(h_{1})(g_{2}f_{1}, g_{2}f_{2}) = \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1}((g_{2}f_{1})(g), (g_{2}f_{2})(g))

= \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1}(f_{1}(gg_{2}), f_{2}(gg_{2}))

= \frac{1}{|G_{1}|} ∑_{g′∈G_{2}} h_{1}(f_{1}(g′), f_{2}(g′))
= coind_{f}(h_{1})(f_{1}, f_{2}).

b) Let G_{2} = ⋃_{i∈I} Im(f)g_{i} be a decomposition of G_{2} into (left) cosets of Im(f)\G_{2}. Then by the G_{1}-equivariance of h_{1} we have

coind_{f}(h_{1})(f_{1}, f_{2}) = \frac{1}{|Ker(f)|} ∑_{i∈I} h_{1}(f_{1}(g_{i}), f_{2}(g_{i})). (4.27)

The coinduced modules are related to the induced RG-modules Ind_{f}(M, h) and to the invariant modules H^{0}(G, (M, h)) in the following way.

Proposition 4.12 a) If f : G_{1} → G_{2} is injective, then there is an isometry of RG_{2}-modules: Coind_{f}(M_{1}, h_{1}) ≃ Ind_{f}(M_{1}, h_{1}). (4.28)

b) If f : G_{1} → G_{2} is surjective, then there is an isometry of RG_{2}-modules:

Coind_{f}(M_{1}, h_{1}) ≃ H^{0}(Ker(f), (M_{1}, h_{1})). (4.29)

Proof. a) Fix a system of (right) coset representatives of G_{2}/Im(f), say G_{2} = ⋃_{i∈I} g_{i}Im(f), and define

ϕ′ : Hom_{RG_{1}}(RG_{2}, M_{1}) → RG_{2} ⊗_{RG_{1}} M_{1}

by ϕ′(t) = ∑_{i∈I} g_{i} ⊗ t(g_{i}^{−1}). It is easy to see that ϕ′ = ϕ_{f,M_{1}}^{−1} is the inverse of the map constructed in Remark 4.3b) and hence is an isomorphism of RG_{2}-modules. It thus remains to show that ϕ′ is an isometry. If t, s ∈ Hom_{RG_{1}}(RG_{2}, M_{1}) then

ind_{f}(h_{1})(ϕ′(t), ϕ′(s)) = ind_{f}(h_{1}) (∑_{i∈I} g_{i} ⊗ t(g_{i}^{−1}), ∑_{i∈I} g_{i} ⊗ s(g_{i}^{−1}))
= ∑_{i∈I} h_{1}(t(g_{i}^{−1}), s(g_{i}^{−1})) = coind_{f}(h_{1})(t, s)

by (4.13) and by formula (4.27) above since {g_{i}^{−1}}_{i∈I} is system of left coset representatives of Im(f)\G_{2}. Thus ϕ′ is an isometry.

b) Put N = Ker(f), and define

ψ : Hom_{RG_{1}}(RG_{2}, M_{1}) → Inv_{N}(M_{1}) = H^{0}(N, M_{1})

by ψ(t) = t(1). If we view Inv_{N}(M_{1}) as a G_{2}-module via the rule f(g_{1})m = g_{1}m for g_{1} ∈ G_{1} and m ∈ Inv_{N}(M_{1}), then ψ is an RG_{2}-module homomorphism because ψ(f(g_{1})t) = (f(g_{1})t)(1) = t(f(g_{1})) = g_{1}t(1) = g_{1}ψ(t) = f(g_{1}) · ψ(t).

It is immediate that ψ is injective because if ψ(t) = 0, then t(1) = 0 and then t(f(g_{1})) = g_{1}t(1) = 0, so t = 0. Moreover, ψ is surjective: if m ∈ Inv_{N}(M_{1}), then define t ∈ Hom_{R}(RG_{2}, M_{1}) by t(f(g_{1})) = g_{1}m. Note that since M is Ninvariant, t is well-defined. Now t(f(g_{1})f(g′_{1})) = g_{1}g′_{1}m = g_{1}t(f(g′_{1})), so t ∈ Hom_{RG_{1}}(RG_{2}, M_{1}), and hence ψ is surjective.

It remains to prove that ψ is an isometry. If s, t ∈ Hom_{RG_{1}}(RG_{2}, M_{1}), then

inv_{N}(h_{1})(ψ(s), ψ(t)) = \frac{1}{|N|}h_{1}(ψ(s), ψ(t)) = \frac{1}{|N|}h_{1}(s(1), t(1)) = coind_{f}(h_{1})(s, t)

by formula (4.27), and so ψ is an isometry.

Just like induction, the process of coinduction enjoys the following functoriality property.

Proposition 4.13 Let f_{1} : G_{1} → G_{2} and f_{2} : G_{2} → G_{3} be two group homomorphisms, and let (M_{1}, h_{1}) be an hermitian RG_{1}-module. Then there is an isometry of hermitian RG_{3}-modules:

Coind_{f_{2}}(Coind_{f_{1}}(M_{1}, h_{1})) ≃ Coind_{f_{2}∘f_{1}}(M_{1}, h_{1}).

Proof. Define

σ : Hom_{RG_{2}}(RG_{3}, Hom_{RG_{1}}(RG_{2}, M_{1})) → Hom_{RG_{1}}(RG_{3}, M_{1})

by σ(t)(g_{3}) = t(g_{3})(1). It is easy to see that σ is an isomorphism of RG_{3}-modules; cf. Cartan-Eilenberg[CE], pp. 28-29, using the identification RG_{3} ⊗_{RG_{2}} RG_{2} ≃ RG_{3}.

To prove that σ is an isometry, let s, t ∈ Hom_{RG_{2}}(RG_{3}, Hom_{RG_{1}}(RG_{2}, M_{1})). Then the RG_{2}-linearity means that t(f_{2}(g′_{2})g_{3}) = g′_{2}t(g_{3}), if g′_{2} ∈ G_{2} and g_{3} ∈ RG_{3}, so

t(f_{2}(g′_{2})g_{3})(g_{2}) = g′_{2}t(g_{3})(g_{2}) = t(g_{3})(g_{2}g′_{2}),

and similarly for s in place of t. Thus we have

coind_{f_{2}}(coind_{f_{1}}(h_{1}))(s, t) = \frac{1}{|G_{2}|} ∑_{g_{3}∈G_{3}} coind_{f_{1}}(h_{1})(s(g_{3}), t(g_{3}))

= \frac{1}{|G_{2}|} ∑_{g_{3}∈G_{3}} \frac{1}{|G_{1}|} ∑_{g_{2}∈G_{2}} h_{1}(s(g_{3})(g_{2}), t(g_{3})(g_{2}))

= \frac{1}{|G_{2}|} ∑_{g_{3}∈G_{3}} \frac{1}{|G_{1}|} ∑_{g_{2}∈G_{2}} h_{1}(s(f_{2}(g_{2})g_{3})(1), t(f_{2}(g_{2})g_{3})(1))

= \frac{1}{|G_{2}|} ∑_{g_{2}∈G_{2}} \frac{1}{|G_{1}|} ∑_{g_{3}∈G_{3}} h_{1}(s(f_{2}(g_{2})g_{3})(1), t(f_{2}(g_{2})g_{3})(1))

= \frac{1}{|G_{2}|} ∑_{g_{2}∈G_{2}} \frac{1}{|G_{1}|} ∑_{g′_{3}∈G_{3}} h_{1}(s(g′_{3})(1), t(g′_{3})(1))

= \frac{1}{|G_{1}|} ∑_{g_{3}∈G_{3}} h_{1}(s(g_{3})(1), t(g_{3})(1))
= coind_{f_{2}∘f_{1}}(h_{1})(σ(s), σ(t)),

which proves that σ is an isometry.

Corollary 4.14 If f : G_{1} → G_{2} is a group homomorphism and (M_{1}, h_{1}) is a hermitian RG_{1}-module, then there is an isometry

H^{0}(G_{2}, Coind_{f}(M_{1}, h_{1})) ≃ H^{0}(G_{1}, (M_{1}, h_{1})).

Furthermore, if f is injective then we also have an isometry

H^{0}(G_{2}, Ind_{f}(M_{1}, h_{1})) ≃ H^{0}(G_{1}, (M_{1}, h_{1})).

Proof. The first assertion follows by taking f_{1} = f and G_{3} = {1} in Proposition 4.13 (cf. proof of Corollary 4.7), and the second follows from the first by Proposition 4.12a).

As we shall see below, the following result will be useful when working with permutation modules:

Corollary 4.15 Let f : G_{1} → G_{2} be a group homomorphism and let X ≤ G_{1} be a subgroup of G_{1}. If k = |X ∩ Ker(f)|^{−1} · 1_{R}, then there is an isometry between the coinduced hermitian permutation module attached to G_{1}/X and the k-twist of the hermitian permutation module attached to G_{2}/f(X):

Coind_{f}(R[G_{1}/X], h_{G_{1}/X}) ≃ (R[G_{2}/f(X)], h_{G_{2}/f(X)})(k) (4.30)

Proof. Let g = f_{|X} : X → Y = f(X) denote the restriction of f to X. Since f ∘ i_{X} = i_{Y} ∘ g, where i_{X} : X → G_{1} and i_{Y} : Y → G_{2} denote the inclusion maps, we obtain by Proposition 4.13 an isometry

Coind_{f}(Coind_{X}^{G_{1}}(R, id)) ≃ Coind_{Y}^{G_{2}}(Coind_{g}(R, id)). (4.31)

By Proposition 4.12a) and Example 4.4b) we have on the one hand that

Coind_{X}^{G_{1}}(R, id) ≃ Ind_{X}^{G_{1}}(R, id) = (R[G_{1}/X], h_{G_{1}/X}); (4.32)

on the other hand we obtain from Proposition 4.12b) that

Coind_{g}(R, id) = H^{0}(Ker(g), (R, id)) = (R, id)(k),

because |Ker(g)| = \frac{1}{k}. It thus follows that

Coind_{Y}^{G_{2}}(Coind_{g}(R, id)) ≃ Ind_{Y}^{G_{2}}(R, id)(k) = (R[G_{2}/f(X)], h_{G_{2}/f(X)})(k).

Substituting this and (4.32) in (4.31) yields the isometry (4.30).

4.4. Frobenius Reciprocity. We are now ready to prove the following version of Frobenius Reciprocity for hermitian RG-modules.

Theorem 4.16 (Frobenius Reciprocity) Let X ≤ G be a subgroup of G and let (M′, h′) be a hermitian RX-module. Then for any hermitian RG-module (M, h) there is an isometry of bilinear R-modules

H^{0}(G, Ind_{X}^{G}(M′, h′) ⊗ (M, h)) ≃ H^{0}(X, (M′, h′) ⊗ Res_{X}^{G}(M, h)).

Proof. By Proposition 4.5 and Corollary 4.14 we have

H^{0}(G, Ind_{X}^{G}(M′, h′) ⊗ (M, h)) ≃ H^{0}(G, Ind_{X}^{G}((M′, h′) ⊗ Res_{X}^{G}(M, h)))
≃ H^{0}(X, (M′, h′) ⊗ Res_{X}^{G}(M, h)).

Corollary 4.17 If S is a finite G-set and (M, h) is a hermitian RG-module, then we have an R-isometry

H^{0}(G, (R[S], h_{S}) ⊗ (M, h)) ≃ ⨁_{s∈G\S} H^{0}(G_{s}, (M, h)),

where G_{s} = {g ∈ G : gs = s} denotes the stabilizer subgroup of s ∈ S.

Proof. By additivity it is enough to consider the case S = G/X, where X ≤ G is a subgroup. By Example 4.4b) we have in this case (R[S], h_{S}) = Ind_{X}^{G}(R, id), and so the assertion follows by taking (M′, h′) = (R, id) in Theorem 4.14.

For hermitian RG-lattices, there is a more general Frobenius reciprocity theorem available. This is based on the following analogue of Proposition 4.5:

Proposition 4.18 Let f : G_{1} → G_{2} be a group homomorphism, and let (M_{i}, h_{i}) be a hermitian RG_{i}-module for i = 1, 2. If M_{2} is R-torsionfree (i.e. an RG_{2}- lattice) or if |Ker(f)| is invertible in R, then there is a canonical isometry

Coind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2}) ≃ Coind_{f}((M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})). (4.33)

In order to prove this proposition it will be convenient to isolate the construction of the desired RG_{2}-module homomorphism which is compatible with the hermitian structures. Subsequently we shall investigate when it is an isometry.

Lemma 4.19 If f : G_{1} → G_{2} is a group homomorphism and (M_{i}, h_{i}) is a hermitian RG_{i}-module for i = 1, 2, then there is a unique RG_{2}-module homomorphism

ν : Hom_{RG_{1}}(RG_{2}, M_{1}) ⊗_{R} M_{2} → Hom_{RG_{1}}(RG_{2}, M_{1} ⊗_{R} M_{2})

such that for t ∈ M := Hom_{RG_{1}}(R_{2}, M_{1}), g_{2} ∈ G_{2}, m_{2} ∈ M_{2}, we have

ν(t ⊗ m_{2})(g_{2}) = t(g_{2}) ⊗ g_{2}m_{2}. (4.34)

Moreover, ν is compatible with the hermitian structures is the sense that

ν^{#}(coind_{f}(h_{1} ⊗ h_{2}) = coind_{f}(h_{1}) ⊗ h_{2}. (4.35)

In addition, ν is an isomorphism if and only if the “cup-product” map

u_{M} : M^{G_{1}} ⊗_{R} M_{2}^{triv} → (M ⊗ M_{2}^{triv})^{G_{1}} (4.36)

is an isomorphism where M_{2}^{triv} denotes the R-module M_{2} endowed with the trivial G_{1}-module structure and M = Hom_{R}(RG_{2}, M_{1}) is viewed as a G_{1}-module in the usual way.

Proof. First note that since ν is R-bilinear in t and m_{2}, the above rule (4.34) defines a unique R-homomorphism

ν : Hom_{RG_{1}}(RG_{2}, M_{1}) ⊗_{R} M_{2} → Hom_{R}(RG_{2}, M_{1} ⊗_{R} M_{2}).

Next we observe that ν(t ⊗ m_{2}) ∈ Hom_{R}(RG_{2}, M_{1} ⊗ M_{2}) is G_{1}-linear because

\begin{aligned}g_{1} · ν(t ⊗ m_{2})(g_{2}) & = & g_{1}(t(g_{2}) ⊗ g_{2}m_{2}) & = & g_{1}t(g_{2}) ⊗ f(g_{1})g_{2}m_{2} \\ & = & t(f(g_{1})g_{2}) ⊗ f(g_{1})g_{2}m_{2} & = & ν(t ⊗ m_{2})(f(g_{1})g_{2}), \end{aligned}

and so Im(ν) ⊂ Hom_{RG_{1}}(RG_{2}, M_{1} ⊗ M_{2}). Moreover, ν is G_{2}-linear because

ν(g_{2}(t ⊗ m_{2}))(g′_{2}) = ν((g_{2}t) ⊗ (g_{2}m_{2}))(g′_{2}) = (g_{2}t)(g′_{2}) ⊗ g′_{2}g_{2}m_{2}
= t(g′_{2}g_{2}) ⊗ g′_{2}g_{2}m_{2}  = ν(t ⊗ m_{2})(g′_{2}g_{2})
= g_{2} · ν(t ⊗ m_{2})(g′_{2}).

We have thus verified the existence of the RG_{2}-linear map ν. The next step is to show that equation (4.35) is valid. For this, let m_{2}, m′_{2} ∈ M_{2} and t, t′ ∈ Hom_{RG_{1}}(RG_{2}, M_{1}). Then

coind_{f}(h_{1} ⊗ h_{2})(ν(t ⊗ m_{2}), ν(t′ ⊗ m′_{2}))

= \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1} ⊗ h_{2}(ν(t ⊗ m_{2})(g), ν(t′ ⊗ m′_{2})(g))

= \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1} ⊗ h_{2}(t(g) ⊗ gm_{2}, t′(g) ⊗ gm′_{2})

= \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1}(t(g), t′(g))h_{2}(gm_{2}, gm′_{2})

= \frac{1}{|G_{1}|} ∑_{g∈G_{2}} h_{1}(t(g), t′(g))h_{2}(m_{2}, m′_{2})
= coind_{f}(h_{1})(t, t′)h_{2}(m_{2}, m′_{2})
= (coind_{f}(h_{1}) ⊗ h_{2})(t ⊗ m_{2}, t′ ⊗ m′_{2}).

It remains to analyze the bijectivity of ν. For this, consider the R-linear map

ν_{1} : M ⊗ M_{2} → Hom_{R}(RG_{2}, M_{1} ⊗ M_{2})

defined by (4.34). We note that ν_{1} is clearly an isomorphism of R-modules since RG_{2} is a free R-module and multiplication by g_{2} is an R-automorphism of M_{2}. Next we observe that

g_{1} · ν_{1}(t ⊗ m_{2})(g_{2}) = g_{1}ν_{1}(t ⊗ m_{2})(f(g_{1})^{−1}g_{2})
= g_{1}(t(f(g_{1})^{−1}g_{2}) ⊗ f(g_{1})^{−1}g_{2}m_{2})
= g_{1}t(f(g_{1})^{−1}g_{2}) ⊗ f(g_{1})f(g_{1})^{−1}g_{2}m_{2}
= (g_{1}t)(g_{2}) ⊗ g_{2}m_{2}
= ν_{1}(g_{1}t ⊗ m_{2})(g_{2}),

which means that ν′ is G_{1}-linear, if we view M_{2} as a trivial G_{1}-module when it appears in M ⊗ M_{2} (but continue to view the M_{2} appearing in M_{1} ⊗ M_{2} as a G_{1}-module via f). Thus ν_{1} is an isomorphism of RG_{1}-modules, and hence induces an R-isomorphism on the invariant spaces:

ν′ = ν_{1}^{G} : (M ⊗M_{2}^{triv})^{G} →^{∼} (Hom_{R}(RG_{2}, M_{1} ⊗M_{2})^{G} = Hom_{RG_{1}}(RG_{2}, M_{1} ⊗M_{2}).

On the other hand, we clearly have the factorization

ν = ν′ ∘ u_{M},

and so it follows that ν is an isomorphism if and only if this is true for u_{M}, as asserted.

Proof of Proposition 4.19. By the above lemma, we only need to verify that ν is an isomorphism under the hypotheses of the proposition.

By the lemma we see that the fact whether or not ν is an isomorphism depends only on the R-module structure of M_{2}, so it is enough to investigate the situation when M_{2} = M_{2}^{triv} has the trivial G_{2}-module structure. In that case the A = RG_{1}- module structure on M_{1} ⊗M_{2} coincides with that of Bourbaki[BA], II.4.2, and also the above map ν coincides with the map ν defined there (cf. loc. cit., equation (7)), if we take B = R, E = RG_{2}, G = M_{1}, F = M_{2} in the notation there. We thus obtain by [BA], II.4.2, Proposition 2, that ν is an isomorphism if G = M_{2} is R-projective (or, equivalently, M_{2} is R-torsionfree since R is a principal ideal domain), or if if RG_{2} is RG_{1}-projective. By the lemma below, this is the case here, and so the assertion follows.

Lemma 4.20 If f : G_{1} → G_{2} is a group homomorphism and R a ring in which |Ker(f)| is invertible, then RG_{2} is a projective RG_{1}-module.

Proof. By factoring f = f_{2} ∘ f_{1} as a surjection followed by an injection, we see that it is enough to consider the case that f is surjective since RG_{2} is a free RG_{1}-module if f is injective. Thus, assume that f is surjective.

To prove that M := RG_{2} is RG_{1}-projective, it is enough to show that M is G_{1}-cohomological trivial, i.e. H^{q}(G, M) = 0 for all q ≥ 1 and all subgroups G ≤ G_{1}. Fix G ≤ G_{1} and put Q = f(G), N = Ker(f) ∩ G. Since |N| is invertible in R, we have H^{q}(N, M) = 0 for q > 0. Thus, by the Hochschild-Serre (spectral) sequence (cf. Serre[Se], VII.6), we have H^{q}(G, M) ≃ H^{q}(Q, M^{N}) = H^{q}(Q, M). But H^{q}(Q, M) = H^{q}(Q, RQ^{n}) = 0, where n = [G_{2} : Q], so the assertion follows.

In view of Proposition 4.12a), the following theorem generalizes the Frobenius Reciprocity Theorem 4.16:

Theorem 4.21 (Generalized Frobenius Reciprocity) Let f : G_{1} → G_{2} be a group homomorphism, and let (M_{i}, h_{i}) be a hermitian RG_{i}-module for i = 1, 2. If M_{2} is an RG_{2}-lattice or if |Ker(f)| is invertible in R, then there is an R-isometry

H^{0}(G_{2}, Coind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2})) ≃ H^{0}(G_{1}, (M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})).

Proof. By Proposition 4.18 and Corollary 4.14 we have

H^{0}(G_{2}, Coind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2}))
≃ H^{0}(G_{2}, Coind_{f}((M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})))
≃ H^{0}(G_{1}, (M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})).

Although we do not need this in the sequel, it would be interesting to know if the restrictive hypotheses (i.e. M_{2} torsionfree or |Ker(f)| invertible in R) in the above theorem can be removed. For our purposes, however, the following strengthening of the theorem to the case that (M_{1}, h_{1}) = (R[S], h_{S}) is a permutation module is sufficient.

Theorem 4.22 (Frobenius Reciprocity for Permutation Modules) Let f : G_{1} → G_{2} be a group homomorphism, and let (M_{1}, h_{1}) = (R[S], h_{S}) be a hermitian RG_{1}-permutation module. Then for any hermitian RG_{2}-module (M_{2}, h_{2}) there is an R-isometry

H^{0}(G_{2}, Coind_{f}(M_{1}, h_{1}) ⊗ (M_{2}, h_{2})) ≃ H^{0}(G_{1}, (M_{1}, h_{1}) ⊗ Res_{f}(M_{2}, h_{2})).

Proof. By additivity it is enough to consider the case S = G_{1}/X, where X ≤ G_{1} is a subgroup. Then by Corollary 4.17 we have

H^{0}(G_{1}, (R[G_{1}/X], h_{G_{1}/X}) ⊗ Res_{f}(M_{2}, h_{2})) ≃ H^{0}(X, Res_{f}(M_{2}, h_{2})). (4.37)

On the other hand, if we put k = |X ∩Ker(f)|^{−1} and Y = f(X), then by Corollary 4.15 and Corollary 4.17 we obtain

H^{0}(G_{2}, Coind_{f}(R[G_{1}/X], h_{G_{1}/X}) ⊗ (M_{2}, h_{2}))
≃ H^{0}(G_{2}, (R[G_{2}/Y], h_{G_{2}/Y})(k) ⊗ (M_{2}, h_{2})) ≃ H^{0}(Y, (M_{2}, h_{2}))(k).

By comparing this with (4.37) we thus see that the assertion follows from the following lemma:

Lemma 4.23 Let f : G_{1} → G_{2} be a group homomorphism and (M_{2}, h_{2}) a hermitian RG_{2}-module. For a subgroup X ≤ G_{1} put Y = f(X) and

k = |Y |/|X| = |X ∩ Ker(f)|^{−1}.

Then we have

H^{0}(X, res_{f}(M_{2}, h_{2})) = H^{0}(Y, (M_{2}, h_{2}))(k).

Proof. Clearly M := Inv_{X}(res_{f}(M_{2})) = Inv_{Y} (M_{2}) as submodules of M_{2}. Thus, if we let h denote the restriction of h_{2} to this submodule, then we have

H^{0}(X, res_{f}(M_{2}, h_{2})) = (M, \frac{1}{|X|}h) = (M, \frac{1}{|Y |}h)(k) = H^{0}(Y, (M_{2}, h_{2}))(k),

and so the lemma follows.

5. The construction of the invariant δ

In this section we shall construct the fundamental invariant δ(M_{1}, M_{2}; M) in three steps. First we introduce the invariant d_{RG,L}(M, h) which is attached to any hermitian RG-module (M, h). This will then be used to construct the preliminary invariant Δ((M_{1}, h_{1}), (M_{2}, h_{2}); M). By introducing the Grothendieck ring H(RG, L) of hermitian RG-modules, this invariant may also be viewed as a pairing of a certain ideal of this ring with the Grothendieck ring Mod(RG)_{non−deg} of non-degenerate RG-modules. Finally we show that Δ induces the desired invariant δ. In the next section we shall study the main properties of this invariant, and then in a subsequent section relate this invariant to the one discussed in the introduction.

5.1. The invariant d_{RG}(M, h). In the previous section (cf. Notation 4.8) we had defined the “0-th cohomology module” H^{0}(G, (M, h)) = Inv_{G}(M, h); taking its discriminant (cf. Definition 3.10) yields the invariant d_{RG,L}(M, h):

Notation 5.1 If (M, h) is an L-valued hermitian RG-module then we let

d_{RG,L}(M, h) = disc_{R}(Inv_{G}(M, h)) = disc_{R}(H^{0}(G, (M, h)))

denote the discriminant of the bilinear module of invariants of (M, h). If the ring R and the field L are clear from the context, then we shall write d_{G} in place of d_{RG,L}.

Remark 5.2 Following the line of thought of section 4, we could also consider the invariant

d′_{RG,L}(M, h) = disc_{R}(M_{G}, h_{G}) = disc_{R}(H_{0}(G, (M, h))).

However, this invariant is of secondary importance because all the applications use only the invariant d_{RG,L}. Moreover, by (4.25) and (4.23) (and Prop. 3.14), these two invariants are related by the formula

(5.1) d′_{RG,L}(M, h) = χ(H_{0}(G, (M, h))_{tor})χ(H^{0}(G, (M, h))_{tor})^{−1}h(G, M)^{2}
·d_{RG,L}(M, h)

where h(G, M) = χ(N̄_{G}) = χ(Ĥ^{0}(G, M)) · χ(Ĥ^{−1}(G, M))^{−1} (5.2)

denotes the “Herbrand quotient” of M. (The latter is a natural generalization of the usual Herbrand quotient which is commonly only defined for cyclic groups G and R = ℤ; cf. Serre[Se], ch. VIII.4.)

We first verify that d_{RG,L}(M, h) ≠ 0 whenever (M, h) is non-degenerate, i.e. whenever disc_{R}(M, h) ≠ 0 (cf. Remark 3.11c)).

Lemma 5.3 Let N′_{G} = |G| − N_{G} = ∑_{g∈G}(1 − g) ∈ RG, where, as before, N_{G} = ∑_{g∈G} g. Then for any hermitian RG-module (M, h) we have

a) M^{G} ⊥ N′_{G}M;

b) M ⊗ K = (M^{G} ⊗ K) ⊕ (N′_{G}M ⊗ K) = (N_{G}M ⊗ K) ⊕ (N′_{G}M ⊗ K);

c) if disc_{R}(M, h) ≠ 0 then also d_{RG}(M, h) ≠ 0. Proof. a) Since (1 − g)m = 0 for all g ∈ G, and m ∈ M^{G}, we have

h(m, N′_{G}m′) = ∑_{g∈G} h(m, (1 − g)m′) = ∑_{g∈G} h((1 − g^{−1})m, m′) = 0,

if m ∈ M^{G} and m′ ∈ M, and so a) holds.

b) Since N_{G}(M ⊗ K) ⊂ (M^{G} ⊗ K), it is enough to prove that N_{G}(M ⊗ K) ⊕ N′_{G}(M ⊗ K) = M ⊗ K and this follows since N_{G} + N′_{G} = n · 1 and n = |G| is invertible in K.

c) Put M′ = \overline{M}^{G} ⊕ N′_{G}\overline{M}. Since the canonical map f : M′ → M is an isogeny by b), it follows from Proposition 3.14 that disc_{R}(M′, f^{#}h) ≠ 0. Furthermore, by a) we have that (M′, f^{#}h) = (\overline{M}^{G}, \overline{h}\overline{M}^{G}) ⊕ (N′_{G}\overline{M}, \overline{h}_{N′_{G}\overline{M}}) and so by Proposition 3.16 we obtain

disc_{R}(M′, f^{#}h) = disc_{R}(\overline{M}^{G}, \overline{h}_{|\overline{M}^{G}})disc_{R}(N′_{G}\overline{M}, \overline{h}_{|N′_{G}\overline{M}}),

which shows that disc_{R}(M^{G}, h_{|M^{G}}) = χ((M^{G})_{tor})^{−1}disc_{R}(\overline{M}^{G}, \overline{h}_{|\overline{M}^{G}}) ≠ 0.

Some elementary properties of the invariant d_{RG} are summarized below. Proposition 5.4 a) The invariant d_{RG} is additive:

d_{RG}((M_{1}, h_{1}) ⊕ (M_{2}, h_{2})) = d_{RG}(M_{1}, h_{1})d_{RG}(M_{2}, h_{2}).

b) In the situation of Proposition 3.12 we have d_{R′G,L′}(M ⊗ R′, h ⊗ R′) = d_{RG,L}(M, h) · R′.

c) The following localization formula is valid:

d_{RG,L}(M, h) = ⋂ d_{R_{𝔭},L}(M_{𝔭}, h_{𝔭}).

d) The invariant of a twist by r ∈ R is given by the formula

d_{RG,L}(M, rh) = r^{rk(H^{0}(G,M))}d_{RG,L}(M, h).

Proof. a) Since H^{0}(G, (M_{1}, h_{1}) ⊕ (M_{2}, h_{2})) = H^{0}(G, (M_{1}, h_{1})) ⊕ H^{0}(G, (M_{2}, h_{2})), this follows directly from Proposition 3.16.

b) Since R′ is flat over R we have H^{0}(G, (M′⊗R′, h⊗R′)) = H^{0}(G, (M, h))⊗R′, and hence the assertion follows from Proposition 3.12.

c) As in the proof of Corollary 3.13, this follows immediately from b).

d) This follows immediately from formula (3.43).

5.2. The invariant Δ. The construction of the invariant Δ is based on the following fundamental but perhaps somewhat technical fact.

Proposition 5.5 Let (M_{i}, h_{i}), i = 1, 2, 3, be three non-degenerate hermitian RGmodules. If M_{1} ⊗K ≃ M_{2} ⊗K, then there is an RG-isogeny f : M_{1} → M_{2} and for any such f there is an automorphism α ∈ Aut_{LG}(M_{1}^{∗}⊗L) such that f^{#}h_{2} = α∘h_{1}. We then have

d_{RG}((M_{1}, h_{1}) ⊗ (M_{3}, h_{3})) = Δ · d_{RG}((M_{2}, h_{2}) ⊗ (M_{3}, h_{3})) (5.3)

where

(5.4)  Δ = χ(inv_{G}(f ⊗ id_{M_{3}}))^{2} det(inv_{G}(α ⊗ id_{M_{3}^{∗}⊗L}))^{−1}
·χ(inv_{G}(M_{1} ⊗ M_{3})_{tor})χ(inv_{G}(M_{2} ⊗ M_{3})_{tor})^{−1}

and inv_{G}(f ⊗ id_{M_{3}}) : inv_{G}(M_{1} ⊗ M_{3}) → inv_{G}(M_{2} ⊗ M_{3}) denotes the induced map on the invariant spaces. In particular, the principal R-module

Δ = d_{G}((M_{1}, h_{1}) ⊗ (M_{3}, h_{3}))d_{G}((M_{2}, h_{2}) ⊗ (M_{3}, h_{3}))^{−1} (5.5)

does not depend on h_{3}.

Proof. We first observe that the last assertion follows immediately from (5.3) since Δ does not depend on h_{3}. (Note that by Lemma 5.3c) it follows that d_{G}((M_{2}, h_{2})⊗ (M, h)) ≠ 0 because (M_{2}, h_{2}) ⊗ (M_{3}, h_{3}) is non-degenerate by Corollary 3.17.)

We now prove the first assertion. By hypothesis there is an isomorphism f̃ ∈ Hom_{KG}(M_{1} ⊗ K, M_{2} ⊗ K) ≃ Hom_{RG}(M_{1}, M_{2}) ⊗ K (cf. [BCA], II.2.7). Thus f̃ = f ⊗ (1/r) with f ∈ Hom_{RG}(M_{1}, M_{2}) and 0 ≠ r ∈ R. Since f : M_{1} → M_{2} is automatically an RG-isogeny, this proves the existence of the desired f.

Next, suppose that an RG-isogeny f is given. Since h_{1}, f^{#}h_{2} : M_{1} → M_{1}^{∗} ⊗ L are non-degenerate, we see by Remark 3.9c) that there is a (unique) α ∈ Aut_{L}(M_{1}^{∗} ⊗ L) such that f^{#}h_{2} = α ∘ h_{1}. Since f^{#}h_{2} and h_{1} are RG-linear, so is α, and so the second assertion follows.

It thus remains to prove equation (5.3). Here we first note that since α ⊗ id is an LG-isomorphism of (M_{1} ⊗ M_{3})^{∗} ⊗ L = (M_{1}^{∗} ⊗ L) ⊗ (M_{3}^{∗} ⊗ L), it induces the isomorphism inv_{G}(α ⊗ id) on the invariant space inv_{G}((M_{1} ⊗ M_{3})^{∗} ⊗ L), and we have inv_{G}(α ⊗ id) ∘ inv_{G}(h_{1} ⊗ h_{3}) = inv_{G}((α ∘ h_{1}) ⊗ h_{3}). Thus by equation (3.42)

d_{G}((M_{1}, h_{1}) ⊗ (M_{3}, h_{3})) = disc(inv_{G}(M_{1} ⊗ M_{3}), inv_{G}(h_{1} ⊗ h_{3}))

= det(inv_{G}(α ⊗ id))^{−1}disc(inv_{G}(M_{1} ⊗ M_{3}), inv_{G}(α ∘ h_{1} ⊗ h_{3}))

= det(inv_{G}(α ⊗ id))^{−1}disc(inv_{G}(M_{1} ⊗ M_{3}), inv_{G}(f^{#}(h_{2}) ⊗ h_{3})).

Furthermore, since inv_{G}(M_{1} ⊗ M_{3} ⊗ K) ≃ inv_{G}(M_{2} ⊗ M_{3} ⊗ K), it follows that the hypothesis of Proposition 3.14 is satisfied, and so we obtain

disc(inv_{G}(M_{1} ⊗ M_{3}), inv_{G}((f^{#}(h_{2})) ⊗ h_{3}))

= disc(inv_{G}(f ⊗ id)^{#}(inv_{G}(M_{2} ⊗ M_{3}), inv_{G}(h_{2} ⊗ h_{3})))

= χ((inv_{G}(M_{1} ⊗ M_{3}))_{tor})χ((inv_{G}(M_{2} ⊗ M_{3}))_{tor})^{−1}
· χ(inv_{G}(f ⊗ id))^{2}disc(inv_{G}(M_{2} ⊗ M_{3}, h_{2} ⊗ h_{3})),

from which formula (5.3) follows.

The following corollary will later play an important role in the construction of the invariant δ.

Corollary 5.6 If, in the situation of Proposition 5.5, we have in addition that M_{1} ≃ M_{2} and that h_{1}, h_{2} are unimodular (i.e. that h_{i} : M_{i} →^{∼} M_{i}^{∗} are isomorphisms), then we have that Δ = R.

Proof. We may assume that M_{1} = M_{2} = M. Then f = id_{M} and α = α_{0} ⊗ L, where α_{0} ∈ Aut(M_{0}), so χ(inv_{G}(id_{M} ⊗ id_{M_{3}})) = R and det(inv_{G}(α ⊗ id_{M_{3}^{∗}⊗L})) = det(inv_{G}(α_{0} ⊗ id_{M_{3}^{∗}})) = R, and hence Δ = R.

We now come to the definition of the invariant Δ, which is defined for RGmodules M which are non-degenerate in the following sense.

Definition 5.7 An RG-module M is non-degenerate if M ⊗ K ≃ M^{∗} ⊗ K (as KG-modules).

Remark 5.8 It is well-known (cf. [Hu], p. 29) that M is non-degenerate if and only if M ⊗ L ≃ M^{∗} ⊗ L, and this is in turn equivalent to the existence of a non-degenerate L-valued hermitian RG-module structure h : M → M^{∗} ⊗ L.

Notation 5.9 Let (M_{1}, h_{1}) and (M_{2}, h_{2}) be two non-degenerate hermitian RGmodules and let M be a non-degenerate RG-module. Then by Remark 5.8 M admits a non-degenerate hermitian RG-module structure h : M → M^{∗} ⊗ L. We now put

Δ_{RG,L}((M_{1}, h_{1}), (M_{2}, h_{2}), M) = d_{G}((M_{1}, h_{1}) ⊗ (M, h))d_{G}((M_{2}, h_{2}) ⊗ (M, h))^{−1},

which, by Proposition 5.5, is independent of the choice of h.

5.3. The Grothendieck ring H(RG, L). In order to be able to work efficiently with the above defined invariant Δ, it advantageous to be able view it as a pairing of certain groups. To this end we introduce here the Grothendieck ring of hermitian RG-modules, which enables us to write everything in a much more compact and transparent form.

Notation 5.10 Let H(RG, L)^{+} denote the set of RG-isometry classes of L-valued non-degenerate hermitian RG-modules (M, h). It is easily checked (using Corollary 3.17) that the operations ⊕ and ⊗ make H(RG, L)^{+} into a commutative semi-ring with unit (R, id_{R}). We let H(RG, L) denote the associated Grothendieck ring (in the sense of Scharlau[Sch], p. 30), and

i_{G} : H(RG, L)^{+} → H(RG, L)

the canonical semi-ring homomorphism. Thus H(RG, L) is a (commutative) ring which is universal with respect to semi-ring homomorphisms f : H(RG, L)^{+} → A to rings A in the sense that f = f′ ∘ i_{G} factors uniquely over a ring homomorphism f′ : H(RG, L) → A. For future reference, note that by the construction of H(RG, L) given in Scharlau[Sch], p. 30, each x ∈ H(RG, L) has the form

x = [M_{1}, h_{1}] − [M_{2}, h_{2}], (5.6)

where [M_{i}, h_{i}] = i_{G}((M_{1}, h_{i})) denotes the image of (the isometry class) of (M_{i}, h_{i}) in H(RG, L).

As a first application of the universal property, we note that there is a canonical ring homomorphism

κ = κ_{G} : H(RG, L) → K_{0}(KG) = G_{0}(KG)

to the Grothendieck ring of (projective) KG-modules (cf. [CR1], p. 406) such that

κ([M, h]) = [M ⊗_{R} K],

for the map (M, h) ↦ [M ⊗ K] defines a semi-ring homomorphism H(RG, L)^{+} → K_{0}(KG) which therefore factors over H(RG, L).

By Remark 5.8 we see that the image of κ is

Im(κ) = K_{0}(KG)_{sym} = {[V_{1}] − [V_{2}] : V_{i}^{∗} ≃ V_{i}},

the subring of K_{0}(KG) generated by the symmetric KG-modules. On the other hand, if we let

H(RG, L)^{0} = Ker(κ)

denote the kernel, then by (5.6) we have

H(RG, L)^{0} = {[M_{1}, h_{1}] − [M_{2}, h_{2}] : M_{1} ⊗ K ≃ M_{2} ⊗ K}. (5.7)

In a similar manner, if Mod(RG) denotes the Grothendieck ring associated to the semi-ring Mod(RG)^{+} consisting of RG-isomorphism classes of (finitely generated) RG-modules, then we have a natural ring homomorphism

ρ = ρ_{G} : H(RG, L) → Mod(RG)

such that ρ([M, h]) = [M]. Note that by Remark (5.8) its image is

Im(ρ) = Mod(RG)_{non−deg},

the subring generated by the non-degenerate RG-modules. Furthermore, the map κ factors over ρ:

κ = \overline{κ} ∘ ρ.

By identifying the group Id(R, L) of non-zero principal R-submodules of L with the quotient group L^{×}/R^{×}, we see by Lemma 5.3c) that d_{RG,L} defines a homomorphism of semi-groups

d_{RG,L}^{+} : H(RG, L)^{+} → Id(R, L) = L^{×}/R^{×}.

By the universal property of Grothendieck groups there is a unique induced group homomorphism

d_{RG} : H(RG, L) → Id(R, L) = L^{×}/R^{×}.

With the above notations we can now express the fundamental invariant Δ as follows.

Theorem 5.11 There is a unique pairing

Δ_{RG,L} : H(RG, L)^{0} × Mod(RG)_{non−deg} → Id(R, L) = L^{×}/R^{×}

such that for any x ∈ H(RG, L)^{0}, y ∈ H(RG, L) we have

Δ_{RG,L}(x, ρ(y)) = d_{RG}(x · y). (5.8)

Proof. Since ρ : H(RG, L) → Mod(RG) is surjective, there exists at most one map satisfying (5.8). To prove existence, we shall use the invariant Δ which was introduced in Notation 5.9. Using the notation of the following Lemma 5.10, we note that Δ defines a bi-additive map

Δ : (H(RG, L)^{+} ⊕ H(RG, L)^{+})^{0} × Mod(RG)_{non−deg}^{+} → L^{×}/R^{×}.

Since it is immediate from the definition (cf. Notation 5.9) that

Δ(M_{1}, h_{1}, M_{1}, h_{1}, M) = 0,

the following lemma shows that Δ induces the desired bi-additive map.

Lemma 5.12 Let A^{+} and B^{+} be two commutative semigroups with associated Grothendieck groups A and B and canonical maps i_{A} : A^{+} → A and i_{B} : B^{+} → B, respectively. Suppose f : A → C is a homomorphism to an abelian group C and put A^{0} = Ker(f) and

(A^{+} ⊕ A^{+})^{0} = {(a_{1}, a_{2}) ∈ A^{+} ⊕ A^{+} : i_{A}(a_{1}) − i_{A}(a_{2}) ∈ A^{0}}.

Furthermore, let D be another abelian group and suppose that

g : (A^{+} ⊕ A^{+})^{0} × B^{+} → D

is a bi-additive map of semigroups such that

g((a, a), b) = 0 (5.9)

44 Ernst Kani for all a ∈ A^{+}, b ∈ B^{+}. Then g induces a unique bi-additive map

\overline{g} : A^{0} × B → D

such that \overline{g}(i_{A}(a_{1}) − i_{A}(a_{2}), i_{B}(b)) = g((a_{1}, a_{2}), b), (5.10) for all (a_{1}, a_{2}) ∈ (A^{+} × A^{+})^{0} and b ∈ B^{+}. Proof. Since A^{0} consists of expressions of the form i_{A}(a_{1}) − i_{A}(a_{2}) and B is generated by the elements i_{B}(b), it is clear that \overline{g} is uniquely determined by (5.10).

To define \overline{g}, fix (a_{1}, a_{2}) ∈ (A^{+} ⊕ A^{+})^{0} and define g_{a_{1},a_{2}} : B^{+} → D by

g_{a_{1},a_{2}}(b) = g((a_{1}, a_{2}), b).

Since g is additive in b, it follows that g_{a_{1},a_{2}} induces a unique homomorphism \overline{g}_{a_{1},a_{2}} : B → D such that

\overline{g}_{a_{1},a_{2}}(i_{B}(b^{+})) = g((a_{1}, a_{2}), b^{+}), for b^{+} ∈ B^{+}.

We note that if (a′_{1}, a′_{2}) ∈ (A^{+} ⊕ A^{+})^{0} is another element, then we have \overline{g}_{a_{1},a_{2}}(b) + \overline{g}_{a′_{1},a′_{2}}(b) = \overline{g}_{a_{1}+a′_{1},a_{2}+a′_{2}}(b), (5.11) for every b ∈ B. Furthermore, it follows from (5.9) that \overline{g}_{a,a} = 0, for all a ∈ A^{+}. (5.12) From this we obtain that

\overline{g}_{a_{1},a_{2}} + \overline{g}_{a_{2},a_{1}} = \overline{g}_{a_{1}+a_{2},a_{1}+a_{2}} = 0,

or, equivalently, that \overline{g}_{a_{2},a_{1}} = −\overline{g}_{a_{1},a_{2}}. (5.13) We next show that if (a_{1}, a_{2}), (a′_{1}, a′_{2}) ∈ (A^{+} ⊕ A^{+})^{0} are such that i_{A}(a_{1}) − i_{A}(a_{2}) = i_{A}(a′_{1}) − i_{A}(a′_{2}), (5.14) then \overline{g}_{a_{1},a_{2}} = \overline{g}_{a′_{1},a′_{2}} (5.15) Indeed, by (5.13) we have i_{A}(a_{1} + a′_{2}) = i_{A}(a_{2} + a′_{1}), so there exits an a_{3} ∈ A^{+} such that a_{1} + a′_{2} + a_{3} = a_{1} + a′_{1} + a_{3}. We thus obtain by (5.13), (5.12) and (5.11) that

\overline{g}_{a_{1},a_{2}} − \overline{g}_{a′_{1},a′_{2}} = \overline{g}_{a_{1},a_{2}} + \overline{g}_{a′_{2},a′_{1}} + \overline{g}_{a_{3},a_{3}} = \overline{g}_{a_{1}+a′_{2}+a_{3},a_{2}+a′_{1}+a_{3}} = 0,

which proves (5.15). Thus, the rule

\overline{g}(a, b) = g_{a_{1},a_{2}}(b), where a = i_{A}(a_{1}) − i_{A}(a_{2}),

defines a map \overline{g} : A^{0} × B → D which is bi-additive by (5.11) and (5.13).

We now investigate some of the properties of the pairing Δ. Its elementary properties are summarized in the next proposition.

Proposition 5.13 a) For any non-degenerate hermitian RG-module (M, h) and any x = [M_{1}, h_{1}] − [M_{2}, h_{2}] ∈ H(RG, L)^{0} we have

d_{G}((M_{1}, h_{1}) ⊗ (M, h)) = Δ(x, M)d_{G}((M_{2}, h_{2}) ⊗ (M, h)). (5.16)

b) If x, y ∈ H(RG, L)^{0} and M, M′ are non-degenerate RG-modules, then

Δ(x, M ⊕ M′) = Δ(x, M) · Δ(x, M′)
Δ(x + y, M) = Δ(x, M) · Δ(y, M)

c) If x ∈ H(RG, L)^{0} and M, M′ are non-degenerate RG-modules, then for any non-degenerate hermitian RG-module structure h on M we have

Δ(x, M ⊗ M′) = Δ(x · [M, h], M′). (5.17)

In particular, Δ(x, M) = Δ(x · [M, h], R). (5.18)

d) Let R′ be a principal ideal domain which is an extension ring of R contained in an extension field L′ of L. Then there is an induced “base change” map

μ : H(RG, L) → H(R′G, L′)

and we have for each non-degenerate RG-module M and x ∈ H(RG, L)^{0} the compatibility relation

Δ_{R′G,L′}(μ(x), M ⊗ R′) = Δ_{RG,L}(x, M) · R′. (5.19)

Proof. a), b) are clear from the construction.

c) Write y = [M, h], z = [M′, h′], where h′ is some non-degenerate hermitian RG-module structure on M′. Then by definition

Δ(x, M ⊗ M′) = d_{RG}(x · (y · z)) = d_{RG}((x · y) · z) = Δ(x · y, M′),

which proves the first assertion. The second follows by taking (M′, h′) = (R, id).

d) Since the map μ^{+}(M, h) = (M ⊗ R′, h ⊗ R′) defines a homomorphism of semi-rings μ^{+} : H(RG, L)^{+} → H(R′G, L′)^{+}, we have by the universal property of Grothendieck rings an induced ring homomorphism μ : H(RG, L) → H(R′G, L′). From Proposition 5.4b) (and (5.16)) it follows that formula (5.19) is valid.

5.4. The pairing δ. We now come to the definition of the invariant δ. For this, it is useful to introduce the following additional Grothendieck rings.

Notation 5.14 Recall from Example 4.1 that a hermitian RG-module (M, h) is called unimodular if h : M →^{∼} M^{∗} is an RG-module isomorphism. Note that in this case M is automatically R-torsionfree, or an RG-lattice in the terminology of Curtis-Reiner[CR1].

Let H(RG)_{uni}^{+} denote the semi-ring consisting of isometry classes of unimodular hermitian RG-modules, and let H(RG)_{uni}^{+} denote the associated Grothendieck ring:

H(RG)_{uni} = {[M_{1}, h_{1}] − [M_{2}, h_{2}] : (M_{i}, h_{i}) is unimodular for i = 1, 2}.

Then by the universal property of Grothendieck rings, the inclusion H(RG)_{uni}^{+} ⊂ H(RG, L)^{+} induces a ring homomorphism

η_{RG,L} : H(RG)_{uni} → H(RG, L).

Furthermore, if we let Mod(RG)_{sym} denote the Grothendieck ring associated to the semi-ring Mod(RG)_{sym}^{+} consisting of isomorphism classes of symmetric (or self-dual) RG-modules M ≃ M^{∗}, then we also have an induced map

ρ_{sym} : H(RG)_{uni} → Mod(RG)_{sym}

which is clearly surjective. We thus have the following commutative diagram:

\begin{matrix}H(RG)_{uni} & →^{η} & H(RG, L) \\ ↓ ^{ρ_{sym}} & & ↓ ^{ρ} & ↘ ^{κ} \\ Mod(RG)_{sym} & →^{\overline{η}} & Mod(RG)_{non−deg} & →^{\overline{κ}} & K_{0}(KG) \end{matrix} (5.20)

Keeping in harmony with earlier notation, we let

Mod(RG)_{sym}^{0} = {[M_{1}] − [M_{2}] ∈ Mod(RG)_{sym} : M_{1} ⊗ K ≃ M_{2} ⊗ K}
= Ker(\overline{κ} ∘ \overline{η}).

Theorem 5.15 There is a unique (bi-additive) pairing

δ_{RG} : Mod(RG)_{sym}^{0} × Mod(RG)_{non−deg} → Id(R, K)

such that δ(ρ_{sym}(x), y) = Δ(η(x), y) (5.21)

for x ∈ Mod(RG)_{sym}^{0} and y ∈ Mod(RG)_{non−deg}.

Proof. Since ρ_{sym} is surjective, the uniqueness assertion is clear. To prove existence, we shall apply Lemma 5.12 again. Here we take A^{+} = Mod(RG)_{sym}^{+}, B^{+} = Mod(RG)_{non−deg}^{+}, f = \overline{κ} ∘ \overline{η} and D = Id(R, K). The map

g : (A^{+} ⊕ A^{+})^{0} → D

is defined by

g((M_{1}, M_{2}), M) = Δ([M_{1}, h_{1}] − [M_{2}, h_{2}], M),

where h_{i} : M_{i} →^{∼} M_{i}^{∗} is any unimodular hermitian RG-module structure on M_{i}, for i = 1, 2; note that by Corollary 5.6 the right hand side does not depend on the choice of h_{i}. Since g clearly satisfies (5.9), we obtain by the lemma the desired map δ.

Notation 5.16 Let M_{1} ≃ M_{1}^{∗} and M_{2} ≃ M_{2}^{∗} be two symmetric (or self-dual) RG-modules such that M_{1} ⊗ K ≃ M_{2} ⊗ K, and let M be a non- degenerate RGmodule (i.e. M ⊗ K ≃ M^{∗} ⊗ K). Then x = [M_{1}] − [M_{2}] ∈ Mod(RG)_{sym}^{0} and y = [M] ∈ Mod(RG)_{non−deg}, and hence we can define the fundamental invariant

δ(M_{1}, M_{2}; M) = δ(x, y), (5.22)

whose main properties will be studied in the next section.

Remark 5.17 Although there is no easy method for actually computing the invariant δ(M_{1}, M_{2}; M), Proposition 5.5 does give us an explicit formula which turns out to be useful in many cases. The drawback is that the formula depends a number of choices, as follows.

To be precise, there are three (independent) choices to be made: we need to choose a unimodular structure h_{i} : M_{i} →^{∼} M_{i}^{∗} on M_{i} for i = 1, 2 and also an RGisogeny f : M_{1} → M_{2}, whose existence is guaranteed by Proposition 5.5. Once these choices have been made, there is a unique α ∈ Aut_{KG}(M_{1}^{∗} ⊗ K) such that f^{#}h_{2} = α ∘ h_{1}, and for any (non-degenerate) M the associated δ-invariant is given by the formula

(5.23) δ(M_{1}, M_{2}; M) = χ(inv_{G}(f ⊗ id_{M}))^{2} det(inv_{G}(α ⊗ id_{M^{∗}⊗K}))^{−1}
·χ(inv_{G}(M_{1} ⊗ M_{tor}))χ(inv_{G}(M_{2} ⊗ M_{tor}))^{−1};

this follows from Proposition 5.5 and equation (5.21), combined with the fact that

(inv_{G}(M_{i} ⊗ M))_{tor} = inv_{G}((M_{i} ⊗ M)_{tor}) = inv_{G}(M_{i} ⊗ M_{tor}),

because M_{i} is torsionfree.

6. The main properties of invariant δ

In this section we shall derive the main properties of the fundamental invariant δ(M_{1}, M_{2}; M) which was constructed in the previous section. For the most part, these properties are generalizations of the main theorems presented in section 2: the theorem on discriminant relations (Theorem 2.5), the Uniform Boundedness Theorem (Theorem 2.8), the theorems on induction and inflation (Theorems 2.12 and 2.14), behaviour with respect to exact sequences (Theorem 2.10), and the Triviality Theorem (Theorem 2.9).

6.1. Elementary properties. Recall from the previous section that the invariant δ(M_{1}, M_{2}; M) was constructed via the pairing

δ : Mod(RG)_{sym}^{0} × Mod(RG)_{non−deg} → Id(R, K),

where the Grothendieck rings Mod(RG)_{sym} and Mod(RG)_{non−deg} are as defined in Notations 5.14 and 5.10, respectively, and Mod(RG)_{sym}^{0} denotes the kernel of the map \overline{κ} ∘ \overline{η} : Mod(RG)_{sym} → K_{0}(KG) (cf. Notation 5.14). Furthermore, Id(R, K) denotes as in Notation 5.10 the group of fractional R-ideals. Then we have:

Theorem 6.1 (Discriminant Relations) Let M_{1} and M_{2} be two symmetric (or self-dual) RG-modules with M_{1} ⊗ K ≃ M_{2} ⊗ K, and let M be a non-degenerate RG-module (i.e. M ⊗ K ≃ M^{∗} ⊗ K). Then x = [M_{1}] − [M_{2}] ∈ Mod(RG)_{sym}^{0} and y = [M] ∈ Mod(RG)_{non−deg}, and for the invariant δ(M_{1}, M_{2}; M) = δ(x, y) the following properties are valid:

a) (Discriminant relation) For any unimodular hermitian RG-module structure h_{i} on M_{i} (i = 1, 2) and any L-valued hermitian RG-module structure h on M we have

disc(Inv_{G}(M_{1} ⊗ M, h_{1} ⊗ h)) = δ(x, y) · disc(Inv_{G}(M_{2} ⊗ M, h_{2} ⊗ h)). (6.1)

b) (Additivity) δ is bi-additive: if M′_{1}, M′_{2} and M′ satisfy the same hypotheses as M_{1}, M_{2} and M respectively, then we have:

δ(M_{1} ⊕ M′_{1}, M_{2} ⊕ M′_{2}; M) = δ(M_{1}, M_{2}; M) · δ(M′_{1}, M′_{2}; M);
δ(M_{1}, M_{2}; M ⊕ M′) = δ(M_{1}, M_{2}; M) · δ(M_{1}, M_{2}; M′).

c) (Symmetry) If M_{3} ≃ (M_{3})^{∗} is symmetric then

δ(M_{1}, M_{2}; M_{3} ⊗ M) = δ(M_{1} ⊗ M_{3}, M_{2} ⊗ M_{3}; M). (6.2)

d) (Base change) If R′ ⊃ R is a principal ideal domain which contains R then

we have δ_{R′}(M_{1} ⊗ R′, M_{2} ⊗ R′; M ⊗ R′) = δ(M_{1}, M_{2}; M) · R′. (6.3)

e) (Localization) We have

δ_{R}(M_{1}, M_{2}; M) = ⋂_{𝔭} δ_{R_{𝔭}}((M_{1})_{𝔭}, (M_{2})_{𝔭}; M_{𝔭}). (6.4)

In particular, if M_{1} and M_{2} are in the same genus class (i.e. (M_{1})_{𝔭} ≃ (M_{2})_{𝔭}, for all 𝔭 ∈ Spec(R)), then δ(M_{1}, M_{2}; M) = R. (6.5)

Similarly, if M′ is in the same genus class as M then

δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; M′). (6.6)

f) (Support) Let n = |G|. Then

δ(M_{1}, M_{2}; M) · R[\frac{1}{n}] = R[\frac{1}{n}]. (6.7)

Thus δ(M_{1}, M_{2}; M) is supported on the prime ideals dividing n.

Proof. a) This follows immediately from the construction of δ, in particular from equations (5.21) and (5.16).

b) δ is bi-additive by construction.

c) Write z = [M_{3}] ∈ Mod(RG)_{sym}. Then δ(x · z, y) = δ(x, \overline{η}(z) · y) by associativity of the tensor product, which proves (6.2).

d) In view of (5.21), this follows directly from Proposition 5.13d).

e) As in the proof of Corollary 3.13, this follows immediately from d).

f) By d) we may assume that R = R[\frac{1}{n}]. Since by hypothesis M_{1}⊗K ≃ M_{2}⊗K, it follows from Curtis-Reiner[CR1], p. 642-643 (and p. 582) that M_{1} and M_{2} are in the same genus class, and so the assertion follows from e).

Remark 6.2 a) In the sequel it will be convenient to have a name for RG-modules M_{1}, M_{2} satisfying the hypotheses of Theorem 6.1. Thus, we shall call a pair (M_{1}, M_{2}) of RG-modules admissible if each M_{i} is symmetric (i.e. M_{i} ≃ M_{i}^{∗}) and if M_{1} ⊗ K ≃ M_{2} ⊗ K.

b) The fact that the invariant δ(M_{1}, M_{2}; M) is defined via the pairing δ on the Grothendieck rings implies a number of other relations as well; for example, if (M_{1}, M_{2}) and (M_{2}, M_{3}) are admissible pairs, then so is (M_{1}, M_{3}) and we have

δ(M_{1}, M_{2}; M)δ(M_{2}, M_{3}; M) = δ(M_{1}, M_{3}; M). (6.8)

As was mentioned in the introduction, most of the interesting examples of RG-modules M for which we want to compute δ(M) are those involving torsion. Since RG-lattices are technically easier to handle, it is therefore often useful to know the relation between the invariant δ(M_{1}, M_{2}; M) attached to M and between the invariant δ(M_{1}, M_{2}; \overline{M}) attached to the torsionfree part \overline{M} = M/M_{tor} of M. While no simple relation seems to exist, we do have the following result:

Proposition 6.3 Let M_{1}, M_{2} and M be as above. Then we have

δ(M_{1}, M_{2}; M) = c_{RG}(M_{1}, M)c_{RG}(M_{2}, M)^{−1}δ(M_{1}, M_{2}; \overline{M}), (6.9)

where for i = 1, 2 the correction term c_{RG}(M_{i}, M) is defined by

(6.10) c_{RG}(M_{i}, M) = χ(inv_{G}(M_{i} ⊗ M_{tor}))χ(inv_{G}(id_{M_{i}} ⊗ π_{M}))^{2}
= χ(inv_{G}(M_{i} ⊗ M_{tor}))^{−1}χ(Coker(inv_{G}(id_{M_{i}} ⊗ π_{M})))^{2}.

Proof. We shall apply formula (5.23) of Remark 5.17. With the notation there we obtain

δ(M_{1}, M_{2}; M)δ(M_{1}, M_{2}; \overline{M})^{−1} = χ(inv_{G}(f ⊗ id_{M}))^{2}χ(inv_{G}(f ⊗ id\overline{M}))^{−2}
·χ(inv_{G}(M_{1} ⊗ M_{tor}))χ(inv_{G}(M_{2} ⊗ M_{tor}))^{−1}
= χ(inv_{G}(id_{M_{1}} ⊗ π_{M}))^{2}χ(inv_{G}(id_{M_{2}} ⊗ π_{M}))^{−2}
·χ(inv_{G}(M_{1} ⊗ M_{tor}))χ(inv_{G}(M_{2} ⊗ M_{tor}))^{−1}
= c_{RG}(M_{1}, M)c_{RG}(M_{2}, M)^{−1}.

Here the first equation follows from (5.23) together with the fact that \overline{M}^{∗} = M^{∗}, so the two determinant factors cancel each other out, and the second follows from the identity

χ(inv_{G}(id_{M_{2}} ⊗ π_{M}))χ(inv_{G}(f ⊗ id_{M})) = χ(inv_{G}(f ⊗ id\overline{M}))χ(inv_{G}(id_{M_{1}} ⊗ π_{M}))

which in turn follows by applying Proposition 3.2 to the commutativity formula

inv_{G}(id_{M_{2}} ⊗ π_{M}) ∘ inv_{G}(f ⊗ id_{M}) = inv_{G}(f ⊗ id\overline{M}) ∘ inv_{G}(id_{M_{1}} ⊗ π_{M}).

6.2. The Uniform Boundedness Theorem. We now turn to the Uniform Boundedness Theorem which, as was explained in section 2, may be viewed as a quantitative generalization of Brauer’s Finiteness Theorem (Corollary 1.3).

In order to be able to state as precise a statement as possible, we introduce the following interesting invariant.

Notation 6.4 Let M and M′ be two RG-lattices with M ⊗ K ≃ M′ ⊗ K. Then the genus defect of M relative to M′ is defined as the R-ideal

γ_{RG}(M : M′) = ∑_{M′′∈Γ} χ(M/M′′), (6.11)

where the sum extends over the set Γ = Γ(M, M′) = {M′′ ⊂ M : M′′ ∨ M′} of all RG-sublattices M′′ofM which are in the same genus class as M′ (i.e. M′′ ∨ M′ in the notation of [CR1]).

Remark 6.5 It is useful to observe that there always exists an RG-submodule M_{max} ∈ Γ(M, M′) such that

χ(M/M_{max}) = γ(M : M′); (6.12)

any such M_{max} is clearly maximal among the RG-submodules in Γ, but there may be more than one such submodule.

To see that at least one such submodule M_{max} exists, note first that this is clear if R = R_{𝔭} is a local ring (hence a discrete valuation ring) because in that case the R-ideals are linearly ordered. In the general case we thus have for each 𝔭 ∈ Spec(R) an R_{𝔭}G-submodule M′′(𝔭) ⊂ M_{𝔭} with M′′(𝔭) ≃ M′_{𝔭}. Put M′′ = ∩ M′′(𝔭); then M′′_{𝔭} = M′′(𝔭) (cf. [BCA], VII.4.3) and

χ(M/M′′) = ⋂ χ(M_{𝔭}/M′′_{𝔭}) = ⋂ γ_{R_{𝔭}}(M_{𝔭} : M′_{𝔭}) ⊃ γ_{R}(M : M′).

On the other hand, since by construction M′′ ∨ M′, it follows that χ(M/M′′) ⊂ γ(M : M′) and hence we see that (6.12) holds for M_{max} = M′′.

The same argument also shows that we have the localization formula

γ_{RG}(M : M′) = ⋂_{𝔭} γ_{R_{𝔭}G}(M_{𝔭} : M′_{𝔭}). (6.13)

Note that as a special case of (6.12) we see that M and M′ are in the same genus class (i.e. M ∨ M′) if and only if γ(M : M′) = R; this justifies the name “genus defect”.

Theorem 6.6 (Uniform Boundedness) If M_{1} and M_{2} are symmetric RGmodules of rank d with M_{1} ⊗ K ≃ M_{2} ⊗ K, define r ∈ R and N ∈ ℕ by the formulae

rR = n^{2d}γ(M_{2} : M_{1})^{4}, N = (2n + 1)d,

where, as before, n = |G|. Then for any non-degenerate RG-module M the ideals

χ(M_{tor})^{N}r^{rk(M)}δ(M_{1}, M_{2}; M) and χ(M_{tor})^{N}r^{rk(M)}δ(M_{1}, M_{2}; M)^{−1}

are R-integral. In other words, for any prime ideal 𝔭 of R, the 𝔭-exponent of δ(M_{1}, M_{2}; M) is uniformly bounded:

|v_{𝔭}(δ(M_{1}, M_{2}; M))| ≤ rk(M)v_{𝔭}(r) + Nv_{𝔭}(χ(M_{tor})).

Proof. In view of the localization formula (6.13) we may assume that R = R_{𝔭} is local. In this case

γ(M_{2} : M_{1}) = ∑_{f} χ(f),

where the sum extends over all RG-isogenies f : M_{1} → M_{2}. It is thus enough to show that

|v_{𝔭}(δ(M_{1}, M_{2}; M))| ≤ (2n + 1)dv_{𝔭}(χ(M_{tor})) + 2drv_{𝔭}(n) + 4rv_{𝔭}(χ(f)), (6.14)

for every RG-isogeny f : M_{1} → M_{2}, where r = dim(M ⊗ K).

Fix such an isogeny f. To compute v_{𝔭}(δ(M_{1}, M_{2}; M)), we shall use formula (5.23) of Remark 5.17. Thus, if h_{i} : M_{i} →^{∼} M_{i}^{∗} is a unimodular structure on M_{i} for i = 1, 2, then α = f^{#}h_{2} ∘ h_{1}^{−1} = f^{∗} ∘ h_{2} ∘ f ∘ h_{1}^{−1} ∈ End_{RG}(M_{1}), and we have by equations (3.7), (3.11) and (3.19) that

det(α) · R = χ(f^{∗})χ(f) = χ(f)^{2}. (6.15)

Now by Proposition 6.3 and formula (5.23) applied to \overline{M} we have

(6.16) δ(M_{1}, M_{2}; M) = c_{RG}(M_{1}, M)c_{RG}(M_{2}, M)^{−1}δ(M_{1}, M_{2}; \overline{M})
= c_{RG}(M_{1}, M)c_{RG}(M_{2}, M)^{−1}
·χ(inv_{G}(f ⊗ id\overline{M}))^{2} det(inv_{G}(α ⊗ id_{\overline{M}^{∗}⊗K}))^{−1}.

Applying the results from the Lemmata 6.7, 6.9 and 6.10 below we therefore deduce from (6.16) that

|v_{𝔭}(δ(M_{1}, M_{2}; M))| ≤ |v_{𝔭}(c_{RG}(M_{1}, M)) − v_{𝔭}(c_{RG}(M_{2}, M))|
+ 2|v_{𝔭}(χ(inv_{G}(f ⊗ id\overline{M})))| + |v_{𝔭}(det(inv_{G}(α ⊗ id_{\overline{M}^{∗}⊗K})))|
≤ (2n + 1)dv_{𝔭}(χ(M_{tor})) + 2drv_{𝔭}(n) + 2rv_{𝔭}(χ(f)) + rv_{𝔭}(det(α))
= (2n + 1)dv_{𝔭}(χ(M_{tor})) + 2drv_{𝔭}(n) + 4rv_{𝔭}(χ(f)),

where the last equality follows from (6.15).

We thus see that the inequality (6.14) and hence Theorem 6.6 are proved once we have established the validity of the following lemmata.

Lemma 6.7 For any prime ideal 𝔭 of R we have

−d · v_{𝔭}(χ(M_{tor})) ≤ v_{𝔭}(c_{RG}(M_{i}, M)) ≤ 2nd · v_{𝔭}(χ(M_{tor})).

Proof. By definition

c_{RG}(M_{i}, M)χ(inv_{G}(M_{i} ⊗ M_{tor})) = χ(Coker(inv_{G}(id_{M_{i}} ⊗ π_{M})))^{2} (6.17)

is R-integral, so we obtain the inequality

v_{𝔭}(c_{RG}(M_{i}, M)) ≥ −v_{𝔭}(χ(inv_{G}(M_{i} ⊗ M_{tor})))
≥ −v_{𝔭}(χ(M_{i} ⊗ M_{tor})) = −dv_{𝔭}(χ(M_{tor})),

since M_{i} ⊗ M_{tor} ≃ M_{tor}^{d} as R-modules. This proves the lower bound.

To prove the upper bound we first note that

Coker(inv_{G}(id_{M_{i}} ⊗ π_{M})) ≃ Im(c) ⊂ H^{1}(G, M_{i} ⊗ M_{tor}), (6.18)

where c denotes the connecting homomorphism in the cohomology sequence

H^{0}(G, M_{i} ⊗ M) → H^{0}(G, M_{i} ⊗ \overline{M}) →^{c} H^{1}(G, M_{i} ⊗ M_{tor})

which results by taking cohomology of the exact sequence

0 → M_{i} ⊗ M_{tor} → M_{i} ⊗ M → M_{i} ⊗ \overline{M} → 0.

Furthermore, for any RG-module T which is R-torsion we have that

χ(H^{1}(G, T)) ⊃ χ(T)^{n} (6.19)

because H^{1}(G, T) is a subquotient of Hom_{R}(RG, T) ≃ T^{n} (as R-modules). Applying this to T = M_{i} ⊗ M_{tor} we obtain from equations (6.17), (6.18) and (6.19) that

v_{𝔭}(c_{RG}(M_{i}, M)) ≤ 2v_{𝔭}(Coker(inv_{G}(id_{M_{i}} ⊗ π_{M})))
≤ 2v_{𝔭}(χ(H^{1}(G, M_{i} ⊗ M_{tor})))
≤ 2n · v_{𝔭}(χ(M_{i} ⊗ M_{tor}))
= 2nd · v_{𝔭}(χ(M_{tor})),

which proves the desired upper bound.

Lemma 6.8 Let u : L_{1} → L_{2} be an isogeny of RG-lattices of rank s, and let inv_{G}(u) : Inv_{G}(L_{1}) → Inv_{G}(L_{2}) denote the induced map on the invariant spaces. Then we have n^{s}χ(u) ⊂ χ(inv_{G}(u)). (6.20)

Proof. Let L′_{i} = N′_{G}L_{i} and L′′_{i} = L_{i}^{G} ⊕ L′_{i}. Since u is RG-linear, it induces Rhomomorphisms u^{G} : L_{1}^{G} → L_{2}^{G}, u′ : L′_{1} → L′_{2} and u′′ : L′′_{1} → L′′_{2}. Now since u′′ = u^{G} ⊕ u′ and nL_{i} ⊂ L′′_{i} ⊂ L_{i} (cf. the proof of Lemma 5.3b)), we see that all these maps are R-isogenies. Thus, by Corollary 3.3 it follows that

χ(u^{G})χ(u′) = χ(u′′) = χ(u)χ(L_{1}/L′′_{1})χ(L_{2}/L′′′_{2})^{−1}.

Since all the χ-ideals are R-integral, we obtain the inclusions

χ(u^{G}) ⊃ χ(u′′) ⊃ χ(u)χ(L_{1}/L′′_{1}).

But χ(L_{1}/L′′_{1}) ⊃ χ(L_{1}/nL_{1}) = n^{s}R, and so the assertion follows.

Corollary 6.9 In the situation of the theorem we have

|v_{𝔭}(χ(inv_{G}(f ⊗ id\overline{M})))| ≤ drv_{𝔭}(n) + rv_{𝔭}(χ(f)).

Proof. Apply Lemma 6.8 to L_{i} = M_{i} ⊗\overline{M} and u = f ⊗id\overline{M}; note that here s = dr. Thus the assertion follows from (6.20) because we have χ(u) = χ(f)^{r}.

Lemma 6.10 Let

ch_{α}(t) = (t − t_{1}) ⋯ (t − t_{d})

denote the characteristic polynomial of α ⊗ K ∈ Aut_{KG}(V), where V = M_{1}^{∗} ⊗ K. Then the characteristic polynomial of α′ = inv_{G}(α ⊗ id_{M^{∗}⊗K}) is of the form

ch_{α′}(t) = (t − t_{1})^{n_{1}} ⋯ (t − t_{d})^{n_{d}},

for suitable integers 0 ≤ n_{k} ≤ r, where 1 ≤ k ≤ d. In particular, det(α′) ∈ R and

det(α′) | det(α)^{r}.

Proof. Write W = M^{∗} ⊗ K. Since dim(W) = r, the characteristic polynomial of α ⊗ id_{W} on V ⊗ W is ch_{α}(t)^{r}. By Lemma 5.3b) we have V ⊗ W = V′ ⊕ V′′ where V′ = Inv_{G}(V ⊗ W) and V′′ = N′_{G}(V ⊗ W). Since α is RG-linear, the map α ⊗ id maps each of the spaces V′ and V′′ into itself. Thus, if α′′ = (α ⊗ id)_{|V′′}, then we obtain

ch_{α′}(t)ch_{α′′}(t) = ch_{α⊗id}(t) = ch_{α}(t)^{r}.

Thus ch_{α′}(t) | ch_{α}(t)^{r}, which proves the first statement. Since α ∈ End(M_{1}), the coefficients of ch_{α}(t) are in R, and hence the same is true for ch_{α′}(t). By Gauss’s lemma it follows that

ch_{α′}(t) | ch_{α}(t)^{r} in R[t];

in particular, the constant coefficient of the former polynomial divides that of the latter, which proves the second statement.

6.3. Frobenius Reciprocity Formulae. We now turn to developing the induction/restriction formalism of the invariant δ. Here we shall first focus on the behaviour of the invariants d_{RG} and Δ, and then later apply this to δ itself. We begin by introducing the following notation.

Notation 6.11 Let f : G_{1} → G_{2} be a group homomorphism. The restriction map

res_{f} : H(RG_{2}, L)^{+} → H(RG_{1}, L)^{+}

as defined in section 4 induces a ring homomorphism

res_{f} : H(RG_{2}, L) → H(RG_{1}, L)

of the associated Grothendieck rings. Furthermore, the induction map

ind_{f} : H(RG_{1}, L)^{+} → H(RG_{2}, L)^{+}

defined in Definition 4.2 induces a homomorphism

ind_{f} : H(RG_{1}, L) → H(RG_{2}, L)

of the associated Grothendieck groups. Similarly, the process of coinduction as defined in Definition 4.10 defines an (additive) homomorphism

coind_{f} : H(RG_{1}, L) → H(RG_{2}, L).

In a similar manner we have a ring homomorphism

res_{f} : Mod(RG_{2})_{non−deg} → Mod(RG_{1})_{non−deg}

and group homomorphisms

ind_{f}, coind_{f} : Mod(RG_{1})_{non−deg} → Mod(RG_{2})_{non−deg};

it is clear that these commute with the forget maps ρ_{G_{1}} and ρ_{G_{2}}:

ρ_{G_{1}} ∘ res_{f} = res_{f} ∘ ρ_{G_{2}}, (co)ind_{f} ∘ ρ_{G_{1}} = ρ_{G_{2}} ∘ (co)ind_{f}.

Similarly, we have induced maps

res_{f} : K_{0}(KG_{2}) → K_{0}(KG_{1}), (co)ind_{f} : K_{0}(KG_{1}) → K_{0}(KG_{2}),

which are compatible with the maps κ_{G_{1}} and κ_{G_{2}} defined in Notation 5.10:

res_{f} ∘ κ_{G_{2}} = κ_{G_{1}} ∘ res_{f}, (co)ind_{f} ∘ κ_{G_{1}} = κ_{G_{2}} ∘ (co)ind_{f},

In particular, we see that

res_{f}(H(RG_{2}, L)^{0}) ⊂ H(RG_{1}, L)^{0}, ind_{f}(H(RG_{1}, L)^{0}) ⊂ H(RG_{2}, L)^{0}, (6.21)

and similarly for coind_{f}.

Remark 6.12 Most of the results of section 4 translate into structural properties of the maps res_{f}, ind_{f} and coind_{f}. Specifically, we have:

a) If f : G_{1} → G_{2} is an injective group homomorphism, then by Proposition 4.5 we have the formula

ind_{f}(x · res_{f}(y)) = ind_{f}(x) · y, (6.22)

56 Ernst Kani which means that ind_{f} is a homomorphism of H(RG_{2}, L)-modules, if we view H(RG_{1}, L) as an H(RG_{2}, L)-module via res_{f}. A similar statement holds if we replace H(RG_{i}, L) by Mod(RG_{i})_{non−deg}.

Similarly, by Proposition 4.18 we have that coind_{f} is a module homomorphism if |Ker(f)| is invertible in R. Moreover, the formula coind_{f}(x · res_{f}(y)) = coind_{f}(x) · y, (6.23) is also valid if y ∈ H(RG_{2}, L)_{lat} lies in the subring H(RG_{2}, L)_{lat} generated by the hermitian RG_{2}-lattices.

b) If f_{1} : G_{1} → G_{2} and f_{2} : G_{2} → G_{3} are group homomorphisms, then by Propositions 4.6 and 4.13 we have ind_{f_{2}∘f_{1}} = ind_{f_{2}} ∘ ind_{f_{1}}, coind_{f_{2}∘f_{1}} = coind_{f_{2}} ∘ coind_{f_{1}} (6.24)

c) From Corollaries 4.14 and 4.7 we obtain the relations d_{RG_{1}} = d_{RG_{2}} ∘ coind_{f}; d′_{RG_{1}} = d′_{RG_{2}} ∘ ind_{f}. (6.25) Furthermore, if f is injective, then coind_{f} = ind_{f} (6.26) by Proposition 4.12a), so in this case we also have d_{RG_{1}} = d_{RG_{2}} ∘ ind_{f}. (6.27)

We now formulate the “Frobenius Reciprocity Formulae” for the invariants d_{RG} and Δ.

Proposition 6.13 Let f : G_{1} → G_{2} be a group homomorphism with |Ker(f)| invertible in R. Then we have d_{RG_{2}}(coind_{f}(x_{1}) · x_{2}) = d_{RG_{1}}(x_{1} · res_{f}(x_{2})), (6.28) if x_{i} ∈ H(RG_{i}, L), i = 1, 2. In particular, for x_{i} ∈ H(RG_{i}, L)^{0} and y_{i} ∈ Mod(RG_{i})_{non−deg}, i = 1, 2, we have

(6.29)  Δ_{RG_{2}}(coind_{f}(x_{1}), y_{2}) = Δ_{RG_{1}}(x_{1}, res_{f}(y_{2})),
(6.30)  Δ_{RG_{1}}(res_{f}(x_{2}), y_{1}) = Δ_{RG_{2}}(x_{2}, coind_{f}(y_{1})).

Moreover, if f is injective then these formulae hold with coind_{f} replaced by ind_{f}. Proof. By (6.23) and (6.24) we have

d_{RG_{2}}(coind_{f}(x_{1}) · x_{2}) = d_{RG_{2}}(coind_{f}(x_{1} · res_{f}(x_{2})))
= d_{RG_{1}}(x_{1} · res_{f}(x_{2})),

which proves equation (6.28) and hence by (5.21) also the other two equations (6.29) and (6.30).

Remark 6.14 If |Ker(f)| is not a unit in R, then formulae (6.28) – (6.30) remain valid if we impose certain additional restrictions on the x_{i} and y_{i}. Indeed, the above proof shows that formula (6.28) is still correct as long as x_{2} ∈ H(RG_{2}, L)_{lat} (which was defined in Remark 6.12). Moreover, by appealing to the Reciprocity Theorem 4.22 in place of (6.23) and (6.24), we see that (6.28) is true whenever x_{1} = [R[S]] is the class of a permutation module. It thus follows that if either x_{1} = [R[S]] is of this type or if y_{2} ∈ Lat(RG_{2})_{non−deg}, the subring of Mod(RG)_{non−deg} which is generated by the classes of (non-degenerate) RG-lattices, then formula (6.29) continues to hold, and similarly, formula (6.30) is true provided that x_{2} ∈ H(RG_{i}, L)_{lat}^{0} = H(RG_{i}, L)^{0} ∩ H(RG_{i}, L)_{lat}.

We now apply the above formulae to the invariant δ. Since the restriction map defines a ring homomorphism

res_{f} : H(RG_{2})_{uni} → H(RG_{1})_{uni},

we see that the analogue of formula (6.30) is valid:

Theorem 6.15 (Coinduction) Let f : G_{1} → G_{2} be a group homomorphism, and let x_{2} ∈ Mod(RG_{2})_{sym} and y_{1} ∈ Mod(RG_{1})_{non−deg}. Then:

δ_{RG_{1}}(res_{f}(x_{2}), y_{1}) = δ_{RG_{2}}(x_{2}, coind_{f}(y_{1})). (6.31)

Proof. Choose x̃_{2} ∈ H(RG_{i})_{uni}^{0} such that ρ_{sym}(x̃_{2}) = x_{i}. Then res(x̃_{2}) ∈ H(RG_{1})_{uni}^{0}, and η_{G_{1}}(res_{f}(x̃_{2})) = res_{f}(η_{G_{2}}(x̃_{2})), so we have by (5.21) that

δ_{RG_{1}}(res_{f}(x_{2}), y_{1}) = Δ_{RG_{1}}(res_{f}(η_{G_{2}}(x̃_{2})), y_{1}) = Δ_{RG_{2}}(η_{G_{2}}(x̃_{2}), coind_{f}(y_{1})),

the latter equality resulting from equation (6.30) (which is valid even if |Ker(f)| is not a unit in R because η_{G_{2}}(x̃_{2}) ∈ H(RG_{2}, K)_{lat}; cf. Remark 6.14). Applying (5.21) once more yields the result.

Corollary 6.16 Let H ⊴ G be a normal subgroup of G, and put Q = G/H. If M_{1} and M_{2} are symmetric RQ-modules with M_{1} ⊗ K ≃ M_{2} ⊗ K, then for any non-degenerate RG-module M we have

δ_{RG}(M_{1}, M_{2}; M) = δ_{RQ}(M_{1}, M_{2}; M^{H}) (6.32)

if we view M_{1} and M_{2} as RG-modules and M^{H} = Inv_{H}(M) as an RQ-module.

Proof. Let f : G_{1} = G → G_{2} = G/H denote the projection map. Then Coind_{f}(M) = M^{H} (cf. Proposition 4.12), so the result follows from the theorem.

On the other hand, in general it is not possible to find an analogue of equation (6.30) for the invariant δ. The basic problem here is that the maps ind_{f} and coind_{f} do not necessarily lift to maps between the Grothendieck rings H(RG_{i})_{uni} of unimodular hermitian RG_{i}-modules, as Corollary 4.15 shows. However, we do have:

Proposition 6.17 If f : G_{1} → G_{2} is an injective group homomorphism and (M_{1}, h_{1}) is a unimodular hermitian RG_{1}-module, then the hermitian RG_{2}-module Coind_{f}(M_{1}, h_{1}) is also unimodular.

Proof. Since f is injective, we have that Coind_{f}(M_{1}, h_{1}) ≃ Ind_{f}(M_{1}, h_{1}) by Proposition 4.12a). Now the definition of induction, particularly (4.10), shows that h_{2} = ind_{f}(h_{1}) : M_{2} = RG_{2} ⊗_{RG_{1}} M → M_{2}^{∗} is given by

h_{2} = ρ_{f,M_{1}} ∘ id_{RG_{2}} ⊗ h_{1}.

Since ρ_{f,M_{1}} : Ind_{f}(M_{1}^{∗}) → Ind_{f}(M_{1})^{∗} is an isomorphism (cf. Example 4.4a)) and h_{1} : M_{1} → M_{1}^{∗} is an isomorphism by hypothesis, it follows that h_{2} is also an isomorphism, as asserted.

Thus, if f : G_{1} → G_{2} is injective, then we have by the proposition an induced map

ind_{f} : H(RG_{1})_{uni} → H(RG_{1})_{uni}

such that ind_{f}([M, h]) = [Ind_{f}(M, h)]. Similarly, as is well known, we have a map

ind_{f} : Mod(RG_{1})_{sym} → Mod(RG_{2})_{sym}.

It is clear from the definitions that these maps are compatible with the maps ρ_{sym}, η and \overline{η} (cf. Notation 5.14), and so we obtain as in (6.21) that

ind_{f}(H(RG_{1})_{uni}^{0}) ⊂ H(RG_{2})_{uni}^{0}, ind_{f}(Mod(RG_{1})_{sym}^{0}) ⊂ Mod(RG_{2})_{sym}^{0}. (6.33)

We thus obtain the following induction formulae for the invariant δ:

Theorem 6.18 (Induction) Let f : G_{1} → G_{2} be an injective group homomorphism. Then for x_{i} ∈ Mod(RG_{i})_{sym}^{0} and y_{i} ∈ Mod(RG_{i})_{non−deg} we have

(6.34)  δ_{RG_{2}}(ind_{f}(x_{1}), y_{2}) = δ_{RG_{1}}(x_{1}, res_{f}(y_{2})),
(6.35)  δ_{RG_{1}}(res_{f}(x_{2}), y_{1}) = δ_{RG_{1}}(x_{2}, ind_{f}(y_{1})).

Proof. Since f is injective, we have that ind_{f} = coind_{f} by (6.26), and so the second equation (6.35) is just a restatement of (6.31). Furthermore, in view of Proposition 6.17, a proof analogous to that of Theorem 6.16 yields equation (6.34).

As was already mentioned, it is not possible to extend Theorem 6.18 to arbitrary maps. However, if one restricts attention to permutation modules, then such an extension is indeed valid:

Theorem 6.19 (Coinduction for Permutation Modules) Let f : G_{1} → G_{2} be a group homomorphism. If S_{1} and S_{2} are G_{1}-sets such that K[S_{1}] ≃ K[S_{2}], then (Coind_{f}(R[S_{1}]), Coind_{f}(R[S_{2}])) is an admissible pair of RG_{2}-modules, and for any non-degenerate RG_{2}-module M_{2} we have

δ_{RG_{2}}(Coind_{f}(R[S_{1}]), Coind_{f}(R[S_{2}]); M_{2}) (6.36)

= ε_{f}(S_{1}, M_{2})ε_{f}(S_{2}, M_{2})^{−1}δ_{RG_{1}}(R[S_{1}], R[S_{2}]; Res_{f}(M_{2})),

where for i = 1, 2

ε_{f}(S_{i}, M_{2}) = ∏_{s∈G_{1}\S_{i}} |Ker(f) ∩ (G_{1})_{s}|^{rk(H^{0}(f((G_{1})_{s}),M_{2}))}. (6.37)

Proof. Since

R[S_{i}] ≃ ⨁_{s∈G_{1}\S_{i}} R[G_{1}/(G_{1})_{s}] = ⨁_{X≤G_{1}} R[G_{1}/X]^{n_{i}(X)}, (6.38)

it follows from Corollary 4.15 that

Coind_{f}(R[S_{i}]) ≃ ⨁_{s∈G_{1}\S_{i}} R[G_{2}/f((G_{1})_{s})] = ⨁_{X≤G_{1}} R[G_{2}/f(X)]^{n_{i}(X)}. (6.39)

Thus Coind_{f}(R[S_{i}]) is again a permutation module, and hence is symmetric. Moreover, since tensoring with K commutes with the process of coinduction, we see that Coind_{f}(R[S_{1}])⊗K ≃ Coind_{f}(R[S_{2}])⊗K, which proves the first assertion.

To prove formula (6.36), put n(X) = n_{1}(X) − n_{2}(X), where n_{i}(X) is defined by equation (6.38), and choose a non-degenerate hermitian RG_{2}-module structure h_{2} on M_{2}. Then by the definition of δ we have

δ_{RG_{1}}(R[S_{1}], R[S_{2}]; Res_{f}(M_{2})) (6.40)

= ∏_{X≤G_{1}} d_{RG_{1}}((R[G_{1}/X], h_{G_{1}/X}) ⊗ (Res_{f}(M), h_{2}))^{n(X)}
= ∏_{X≤G_{1}} d_{RG_{2}}(Coind_{f}(R[G_{1}/X], h_{G_{1}/X}) ⊗ (M_{2}, h_{2}))^{n(X)};

here we have used the reciprocity formula (6.28) which is applicable in this case by Remark 6.14.

For a subgroup X ≤ G_{1} put k(X) = |X ∩ Ker(f)|^{−1} and

r(f(X)) = rk(H^{0}(G_{2}, R[G_{2}/f(X)] ⊗ M_{2}])) = rk(H^{0}(f(X), M_{2})),

where the latter equality follows from Corollary 4.17. Now by Corollary 4.15 we have

Coind_{f}(R[G_{1}/X], h_{G_{1}/X}) ≃ (R[G_{2}/f(X)], h_{G_{2}/f(X)})(k(X)),

and so substituting this in the above computation (6.40) we obtain by Proposition 5.4d) that

δ_{RG_{1}}(R[S_{1}], R[S_{2}]; Res_{f}(M_{2}))
= ∏_{X≤G_{1}} d_{RG_{2}}((R[G_{2}/f(X)], h_{G_{2}/f(X)})(k(X)) ⊗ (M_{2}, h_{2})))^{n(X)}
= ∏_{X≤G_{1}}(k(X)^{r(f(X))}d_{RG_{2}}((R[G_{2}/f(X)], h_{G_{2}/f(X)}) ⊗ (M_{2}, h_{2}))))^{n(X)}
= ( ∏_{X≤G_{1}} k(X)^{n(X)r(f(X))})δ_{RG_{2}}(Coind_{f}(R[S_{1}]), Coind_{f}(R[S_{2}]); M_{2}),

where the last equation follows from the definition of δ and (6.39). This proves (6.36) because the factor in front of δ is by (6.39) equal to ε_{f}(S_{1}, M_{2})ε_{f}(S_{2}, M_{2})^{−1}.

6.4. The exact sequence formula. We next investigate the behaviour of the invariant δ(M_{1}, M_{2}; M) with respect to exact sequences. Although δ is in general not additive on exact sequences, the difference can be understood in terms of group cohomology. To this end we first introduce the following notation.

Notation 6.20 Let f : M′ → M′′ be a homomorphism of RG-modules. Then for any RG-module M we have the induced R-module homomorphism

f_{M}^{G} = inv_{G}(id_{M} ⊗ f) : Inv_{G}(M ⊗ M′) → Inv_{G}(M ⊗ M′′)

on the invariant spaces. We put

ψ_{G}(M, f) = χ(Coker(f_{M}^{G})).

Moreover, if N is an RG-module with ψ_{G}(M, f) ≠ 0, then we put

ψ_{G}(M, N; f) = ψ_{G}(M, f)ψ_{G}(N, f)^{−1}.

With this notation we can now formulate the exact sequence formula, which may be viewed as a generalization of Proposition 6.3 which relates the δ-invariants of M, M_{tor} and \overline{M} = M/M_{tor}.

Theorem 6.21 (Exact Sequence Formula) Let

0 → M′ →^{f} M →^{g} M′′ → 0

be an exact sequence of RG-modules. If any two of the modules M, M′ and M′′ are non-degenerate, then so is the third, and for any admissible pair (M_{1}, M_{2}) of RG-modules we have the relation

δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; M′)δ(M_{1}, M_{2}; M′′)ψ(M_{1}, M_{2}; g)^{2}, (6.41)

where ψ(M_{1}, M_{2}; g) = ψ(M_{1}, g)ψ(M_{2}, g)^{−1} is as defined above. Furthermore, for i = 1, 2 we have ψ(M_{i}, g) = χ(Ker((id_{M_{i}} ⊗ f)^{1})), (6.42)

where (id_{M_{i}} ⊗ f)^{1} : H^{1}(G, M_{i} ⊗ M′) → H^{1}(G, M_{i} ⊗ M) denotes the induced map on cohomology.

Proof. Since KG is semi-simple we have M ⊗ K ≃ (M′ ⊗ K) ⊕ (M′′ ⊗ K), and so we see that if M′ ⊗K ≃ (M′)^{∗} ⊗K and M′′ ⊗K ≃ (M′′)^{∗} ⊗K, then also M ⊗K ≃ M^{∗} ⊗ K. On the other hand, if M ⊗ K ≃ M^{∗} ⊗ K and M′ ⊗ K ≃ (M′)^{∗} ⊗ K, then also M′′ ⊗ K ≃ (M′′)^{∗} ⊗ K because the cancellation law is valid for KG-modules.

Since the same argument holds when M′ and M′′ are interchanged, this proves the first assertion.

We next prove equation (6.42). Since M_{i} is R-torsionfree, hence R-flat, the sequence

0 → M_{i} ⊗ M′ → M_{i} ⊗ M → M_{i} ⊗ M′′ → 0

is exact, so by the long exact sequence of cohomology, viz.

(M_{i} ⊗ M)^{G} →^{g_{M_{i}}^{G}} (M_{i} ⊗ M′′)^{G} →^{c} H^{1}(G, M_{i} ⊗ M′) →^{(id⊗f)^{1}} H^{1}(G, M_{i} ⊗ M),

we obtain Coker(g_{M_{i}}^{G}) ≃ Ker((id_{M_{i}} ⊗ f)^{1}), and hence (6.42) follows.

Finally, to prove formula (6.41), let t_{0} : M_{1} → M_{2} be an RG-isogeny, let h_{i} : M_{i} → M_{i}^{∗} be unimodular hermitian structures on M_{i} for i = 1, 2, and put α_{0} = (t_{0}^{#}h_{2} ∘ h_{1}^{−1}) ⊗ K ∈ Aut_{KG}(V_{1}), where V_{1} = M_{1}^{∗} ⊗ K. Furthermore, let

t = inv_{G}(t_{0} ⊗ id_{M}) : (M_{1} ⊗ M)^{G} → (M_{2} ⊗ M)^{G},

and α = inv_{G}(α_{0} ⊗ id_{V}) ∈ Aut_{KG}(V_{1} ⊗ V), where V = M^{∗} ⊗ K, and define similarly t′, α′, V′ (respectively, t′′, α′′, V′′) by replacing M by M′ (respectively by M′′). Then by formula (5.23) we have

(6.43)  d =^{def} δ(M_{1}, M_{2}; M)δ(M_{1}, M_{2}; M′)^{−1}δ(M_{1}, M_{2}; M′′)^{−1}
= χ(t)^{2} det(α)^{−1} · χ(t′)^{−2} det(α′) · χ(t′′)^{−2} det(α′′).

Now since the diagram

\begin{aligned}0 & → & (V_{1} ⊗ V′′)^{G} & →^{(id⊗g^{∗})^{G}} & (V_{1} ⊗ V)^{G} & →^{(id⊗f^{∗})^{G}} & (V_{1} ⊗ V′)^{G} & → & 0 \\ & & ↓ _{α′′} & & ↓ ^{α} & & ↓ _{α′} \\ 0 & → & (V_{1} ⊗ V′′)^{G} & →^{(id⊗g^{∗})^{G}} & (V_{1} ⊗ V)^{G} & →^{(id⊗f^{∗})^{G}} & (V_{1} ⊗ V′)^{G} & → & 0 \end{aligned} (6.44)

is commutative with exact rows, we see that the determinant terms in (6.43) cancel out, and so we are left with

d = χ(t)^{2}χ(t′)^{−2}χ(t′′)^{−2}. (6.45)

On the other hand, since the diagram

\begin{aligned}0 & → & (M_{1} ⊗ M′)^{G} & →^{f_{M_{1}}^{G}} & (M_{1} ⊗ M)^{G} & →^{g_{M_{1}}^{G}} & (M_{1} ⊗ M′′)^{G} \\ & & ↓ _{t′} & & ↓ ^{t} & & ↓ _{t′′} \\ 0 & → & (M_{2} ⊗ M′)^{G} & →^{f_{M_{2}}^{G}} & (M_{2} ⊗ M)^{G} & →^{g_{M_{2}}^{G}} & (M_{2} ⊗ M′′)^{G} \end{aligned} (6.46)

is also commutative with exact rows, we obtain from Proposition 3.4 the relation

χ(t) = χ(t′)χ(s′′),

where s′′ : Im(g_{M_{1}}^{G}) → Im(g_{M_{2}}^{G}) denotes the restriction of t′′ to Im(g_{M_{1}}^{G}). Now by Corollary 3.3 we have

χ(s′′) = χ(t′′)χ(Coker(g_{M_{1}}^{G}))χ(Coker(g_{M_{2}}^{G}))^{−1} = χ(t′′)ψ(M_{1}, M_{2}, g),

which yields

χ(t) = χ(t′)χ(t′′)ψ(M_{1}, M_{2}, g).

Substituting this in (6.45) gives equation (6.41).

The following corollary will be used and strengthened in the next subsection.

Corollary 6.22 If, in the situation of Theorem 6.21, M′ is cohomologically trivial, then we have

δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; M′)δ(M_{1}, M_{2}; M′′). (6.47)

Similarly, if M is cohomologically trivial then we have

(6.48) δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; M′)δ(M_{1}, M_{2}; M′′)
·χ(H^{1}(G, M_{1} ⊗ M′))^{2}χ(H^{1}(G, M_{2} ⊗ M′))^{−2}.

Proof. If M′ is cohomologically trivial, then so is M_{i} ⊗ M′ (cf. Serre[Se], IX.3), and so Ker((id_{M_{i}} ⊗ f)^{1}) = H^{1}(G, M_{i} ⊗ M′) = 0. This proves (6.47), and (6.48) is proved similarly.

6.5. Triviality. The last result which we shall prove here is the following useful formula which justifies in part the unusual definition of the discriminant introduced in section 3:

Theorem 6.23 (Triviality) Let M be a non-degenerate RG-module which is cohomologically trivial. Then the associated δ-invariant

δ(M_{1}, M_{2}; M) = R

is trivial for any pair (M_{1}, M_{2}) of symmetric RG-modules with M_{1} ⊗K ≃ M_{2} ⊗K.

We shall prove this theorem first for projective modules and then for arbitrary cohomologically trivial modules by using the exact sequence formula.

The key to proving the theorem for projective modules is the following technical lemma concerning the map

ρ_{M} = ρ_{f,M} : R ⊗ M^{∗} → (R ⊗ M)^{∗}

which was defined in Remark 4.4; here f : G → {1} denotes the trivial map. More explicitly, ρ_{M} is defined by the formula

ρ_{M}(r ⊗ m^{∗})(r′ ⊗ m′) = rr′m^{∗}(∑_{g∈G} gm′) = rr′m^{∗}(N_{G}m′),

where r, r′ ∈ R, m^{∗} ∈ M^{∗} and m′ ∈ M.

Lemma 6.24 If M is a projective RG-module, then

ρ_{M} : R ⊗_{RG} M^{∗} → (R ⊗ M)^{∗}

is an isomorphism.

Proof. Let us first suppose that M = RG. Then R ⊗_{RG} RG^{∗} = R, and if {g^{∗}}_{g∈G} denotes a dual basis of the standard basis of RG, we have

ρ_{M}(1 ⊗ g^{∗})(1 ⊗ g′) = g^{∗}( ∑_{g_{1}∈G} g_{1}g′) = 1,

for all g, g′ ∈ G. Thus ρ_{M}(1 ⊗ g^{∗}) = 1 ∈ Hom_{R}(R ⊗_{RG} RG, R) = R, for all g ∈ G, which shows that ρ_{M} is an isomorphism in this case. Since ρ_{M} is additive in M, it follows that ρ_{M} is an isomorphism for any direct factor of a free RG-module, which proves the lemma.

Proposition 6.25 If P is a projective RG-module with a unimodular hermitian RG-module structure h : P →^{∼} P^{∗}, then

H_{0}(G, (P, h)) ≃ H^{0}(G, (P, h)) (6.49)

is a unimodular bilinear R-module. In particular, we have that

d_{RG}(P, h) = d′_{RG}(P, h) = R. (6.50)

Proof. We first note that since P is projective we have Ĥ^{−1}(G, P) = Ĥ^{0}(G, P) = 0, and so by the exact sequence (4.23) and Remark 4.9 we obtain the desired isometry (6.49).

From (4.10) we see that the bilinear structure ind(h) on H_{0}(P) = Ind_{f}(P) is given by

ind_{f}(h) = ρ_{f,P} ∘ id_{R} ⊗ h,

where f : G → {1} denotes the trivial map. Now h is an isomorphism by hypothesis, hence so is id ⊗ h, and thus it follows from Lemma 6.24 that ind(h) is also an isomorphism, as asserted.

Corollary 6.26 If (P, h) is as in the proposition, then for any unimodular hermitian RG-module (M, h′) we have

d_{RG}((M, h′) ⊗ (P, h)) = R.

In particular, for any admissible pair (M_{1}, M_{2}) the associated δ-invariant is trivial:

δ(M_{1}, M_{2}; P) = R.

Proof. We first note that P′ = M ⊗ P is again RG-projective. Indeed, it is cohomologically trivial by Serre[Se], IX.3, and hence projective by Serre[Se], IX.5, because P′ is clearly R-torsionfree. Thus, since (P′, h′′) = (P, h)⊗(M, h′) is again unimodular, the first equation follows directly from (6.50).

Moreover, the second equation follows from the first because

δ(M_{1}, M_{2}; P) = d_{RG}((M_{1}, h_{1}) ⊗ (P, h)) · d_{RG}((M_{2}, h_{2}) ⊗ (P, h))^{−1},

for any choice of a unimodular structure h_{i} on M_{i}, for i = 1, 2.

Remark 6.27 In the case that (P, h) = (RG, h_{G}) is the regular hermitian RGmodule, a more direct proof of this corollary can be given via the Frobenius Reciprocity Theorem. Indeed, since (RG, h_{G}) ≃ Ind_{H}^{G}(R, id_{R}), where H = {1} (cf. Example 4.4b)), we obtain by the Frobenius Reciprocity formula (6.28) (together with (6.26)) that

d_{RG}((M, h) ⊗ (RG, h_{G})) ≃ d_{RH}(Res(M, h) ⊗ (R, id_{R})) = disc_{R}(M, h) = R.

We have thus proven the Triviality Theorem for symmetric projective RGmodules. To go further, we shall use:

Lemma 6.28 If R is a discrete valuation ring, then every projective non-degene-rate RG-module P is symmetric.

Proof. Since P is non-degenerate, we have an KG-isomorphism P ⊗ K ≃ P^{∗} ⊗ K. But P^{∗} is also projective (cf. Curtis-Reiner[CR1], (10.29)), so it follows from [CR1], (32.1), that P ≃ P^{∗}, as asserted.

Lemma 6.29 If M is a non-degenerate RG-module which is cohomologically trivial, then M has a presentation

0 → P′ → P → M → 0, (6.51)

where P and P′ are non-degenerate projective RG-modules.

Proof. Let 0 → M′ → P → M → 0 be an RG-module presentation of M with P = RG^{m} a free module. Since M and P are non-degenerate, so is M′ by Theorem 6.21. Furthermore, since M and P are cohomologically trivial, the long exact sequence of cohomology shows that this is also true for M′. Since M′ ⊂ P is R-torsionfree, it follows that M′ = P′ is RG-projective (cf. Serre[Se], IX.5).

Proof of the Triviality Theorem. Suppose first that M = P is a (non-degenerate) projective RG-module. If 𝔭 ∈ Spec(R) is any prime ideal of R, then M_{𝔭} is a symmetric R_{𝔭}G-module by Lemma 6.28, and so δ_{R_{𝔭}G}((M_{1})_{𝔭}, (M_{2})_{𝔭}; M_{𝔭}) = R_{𝔭} by Corollary 6.26. From the localization formula (6.4) we thus obtain that

δ_{RG}(M_{1}, M_{2} :, M) = ⋂_{𝔭} δ_{R_{𝔭}G}((M_{1})_{𝔭}, (M_{2})_{𝔭}; M_{𝔭}) = ⋂_{𝔭} R_{𝔭} = R,

which proves the theorem if M is projective.

Now suppose that M is an arbitrary cohomologically trivial module, and let P and P′ be as in Lemma 6.29. Applying the exact sequence formula, notably Corollary 6.22, to (6.51), we obtain

δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; P)δ(M_{1}, M_{2}; P′)^{−1}.

Since P and P′ are RG-projective, we have δ(M_{1}, M_{2}; P) = δ(M_{1}, M_{2}; P′) = R by the case treated above, and so the theorem is proved.

With the help of the Triviality Theorem, Corollary 6.22 can now be strengthened. It is convenient to separate the two cases of the corollary as follows.

Corollary 6.30 Let

0 → M′ →^{f} M →^{g} M′′ → 0

be an exact sequence of non-degenerate RG-modules, and suppose that M′ is cohomologically trivial. Then for any admissible pair (M_{1}, M_{2}) of RG-modules we have: δ(M_{1}, M_{2}; M) = δ(M_{1}, M_{2}; M′′). (6.52)

Proof. Since δ(M_{1}, M_{2}; M′) = R by the Triviality Theorem, this follows immediately from (6.47).

Corollary 6.31 Let

0 → M′ → P_{r} → ⋯ → P_{1} → M′′ → 0

be an exact sequence of non-degenerate RG-modules, where the P_{i} are cohomologically trivial for 1 ≤ i ≤ r. Fix an admissible pair (M_{1}, M_{2}) of RG-modules and write δ(M′) = δ(M_{1}, M_{2}; M′) and δ(M′′) = δ(M_{1}, M_{2}; M′′). Then we have:

(6.53)  δ(M′) = δ(M′′)^{(−1)^{r}} · χ^{r}(G, M_{1} ⊗ M′)^{2}χ^{r}(G, M_{2} ⊗ M′)^{−2},
= δ(M′′)^{(−1)^{r}} · χ_{r}(G, M_{1} ⊗ M′′)^{2}χ_{r}(G, M_{2} ⊗ M′′)^{−2},

where, for an RG-module M, the truncated Euler characteristics χ^{r}(G, M) and χ_{r}(G, M) are defined by

χ^{r}(G, M) = ∏_{i=1}^{r} χ(H^{i}(G, M))^{(−1)^{i+1}},

χ_{r}(G, M) = ∏_{i=0}^{r−1} χ(Ĥ^{−i}(G, M))^{(−1)^{r−1−i}}.

Proof. We first note that the second equation follows from the first since for i = 1, 2 we have by the usual dimension-shifting argument that

Ĥ^{q}(G, M_{i} ⊗ M′) ≃ Ĥ^{q−r}(G, M_{i} ⊗ M′′), for all q ∈ ℤ;

here we have used the fact that M_{i} ⊗ P_{k} is cohomologically trivial for 1 ≤ k ≤ r.

To prove the first equality in (6.53), we shall induct on r. If r = 1, then the assertion is just equation (6.48) of Corollary 6.22, combined with the fact that δ(P_{1}) = R by the Triviality Theorem; thus, assume r > 1.

Applying the induction hypothesis to the exact sequence

0 → M′ → P_{r} → ⋯ P_{2} → M → 0,

where M = Ker(P_{1} → M′′), yields

δ(M′) = δ(M)^{(−1)^{r−1}} · χ^{r−1}(M_{1} ⊗ M′)^{2}χ^{r−1}(M_{2} ⊗ M′)^{−2},

and similarly, consideration of the exact sequence

0 → M → P_{1} → M′′ → 0

shows that

δ(M) = δ(M′′)^{−1}χ(H^{1}(G, M_{1} ⊗ M))^{2}χ(H^{1}(G, M_{2} ⊗ M))^{−2}.

Thus, by combining the last two equations we obtain

δ(M′) = δ(M′′)^{(−1)^{r}} · c_{1}^{2}c_{2}^{−2}, (6.54)

where c_{i} = χ^{r−1}(M_{i} ⊗ M′)χ(H^{1}(G, M_{i} ⊗ M))^{(−1)^{r−1}}. But by dimension-shifting as above we have

Ĥ^{q}(G, M_{i} ⊗ M) ≃ Ĥ^{q+r−1}(G, M_{i} ⊗ M′),

and so it follows that

c_{i} = χ^{r−1}(M_{i} ⊗ M′)χ(H^{r}(G, M_{i} ⊗ M′))^{(−1)^{r−1}} = χ^{r}(M_{i} ⊗ M′).

Substituting this into (6.54) yields the first equation of (6.53), as desired.

7. The invariant δ and character relations

We now interpret the results obtained in the previous section in terms of character relations. Since character relations are naturally associated with the subgroup module 𝕊(G), we first introduce this space. However, the functorial properties of character relations are best understood in terms of the Burnside ring Ω(G), and so we study this ring in some detail. Next we observe that the δ-invariant may be viewed as a pairing of a certain ideal Ω(G)^{0} ⊂ Ω(G) with the Grothendieck ring Mod(RG)_{non−deg} of non-degenerate RG-modules, and this will then be used to translate the results of the previous section into the form in which they were presented in section 2.

7.1. The space of character relations. Recall from the introduction that a character relation is a relation of the form

∑_{H≤G} n_{H}1_{H}^{∗} = 0, (7.1)

where the n_{H} are integers and 1_{H}^{∗} = Ind_{H}^{G}(1_{H}) denotes the permutation character of the G-set G/H. To study such relations, consider the subgroup module

𝕊(G) = ∑_{H≤G} ℤ · H

which is the free abelian group generated by the subgroups H ≤ G. Thus, each element x ∈ 𝕊(G) may be written as a formal sum x = ∑ n_{H}H, or as a tuple x = {n_{H}}; the latter was the notation used in the introduction.

We have a natural additive map

ch_{G} : 𝕊(G) → ch(ℚG)

from the subgroup space 𝕊(G) to the character ring ch(ℚG) = K_{0}(ℚG) which associates to each subgroup H ≤ G its permutation character:

ch_{G}(H) = 1_{H}^{∗}.

The kernel of the character map ch_{G} is called the group of character relations of G and is denoted by

𝕊(G)^{0} = Ker(ch).

Clearly an element x = ∑ n_{H}H ∈ 𝕊(G) lies in 𝕊(G)^{0} if and only if its coefficients satisfy the character relation (7.1); this justifies the above terminology for 𝕊(G)^{0}.

For our purposes it will be also useful to consider the map

ch_{KG} : 𝕊(G) → ch(KG) = K_{0}(KG)

defined by ch_{KG}(X) = Ind_{X}^{G}(K) for X ≤ G. However, this map is not really something substantially new for we have

ch_{KG} = j_{KG} ∘ ch_{G},

where j_{KG} : K_{0}(ℚG) → K_{0}(KG) denotes the canonical injection. (In the case that char(K) = 0, this is just the base-change map induced by the inclusion ℚ ⊂ K, whereas when char(K) = p ≠ 0, this is the map obtained by the composition K_{0}(ℚG) → K_{0}(ℚ_{p}G) = K_{0}(𝔽_{p}G) → K_{0}(KG).) It thus follows that

𝕊(G)^{0} = Ker(ch_{KG}).

We note that the map ch_{KG} factors over the map

\overline{κ} ∘ \overline{η} : Mod(RG)_{sym} → K_{0}(KG)

defined in Notation 5.14; in fact, if we define the map

ch_{RG} : 𝕊(G) → Mod(RG)_{sym}

by ch_{RG}(X) = [Ind_{X}^{G}(R)] for X ≤ G, then we clearly have

ch_{KG} = \overline{κ} ∘ \overline{η} ∘ ch_{RG}. (7.2)

In particular, we see that

ch_{RG}(𝕊(G)^{0}) ⊂ Mod(RG)_{sym}^{0}. (7.3)

We next want to consider functorial properties of character relations. To this end, let f : G_{1} → G_{2} be group homomorphism. We then have an induced additive map

f_{∗} : 𝕊(G_{1}) → 𝕊(G_{2})

which is defined by the rule f_{∗}(X) = f(X) for X ≤ G_{1}. We note that f_{∗} preserves character relations, viz. f_{∗}(𝕊(G_{1})^{0}) ⊂ 𝕊(G_{2})^{0}, (7.4)

because by Corollary 4.15 we have the formula

coind_{f} ∘ ch_{G_{1}} = ch_{G_{2}} ∘ f_{∗}. (7.5)

It is more difficult, however, to construct a map f^{∗} which relates the character relations of G_{2} to those of G_{1}. Here we cannot work integrally, but have to allow denominators. This means that we can define f^{∗} only as a ℚ-linear map

f^{∗} : 𝕊_{ℚ}(G_{2}) → 𝕊_{ℚ}(G_{1}),

where 𝕊_{ℚ}(G_{i}) = 𝕊(G_{i}) ⊗ ℚ. Explicitly, f^{∗} is given by

f^{∗}(Y) = \frac{1}{|G_{1}|} ∑_{g∈G_{2}} \frac{|f(gY g^{−1})|}{|Y |}f^{−1}(gY g^{−1}) = ∑_{X≤G_{1}} m_{f}(Y, X)X, (7.6)

where m_{f}(Y, X) = #{g ∈ G_{2} : f^{−1}(gY g^{−1}) = X}/([G_{1} : X] · |Y |). However, the fact that denominators appear here limits the usefulness of f^{∗}, particularly since there is a way to circumvent this difficulty.

7.2. Burnside rings. It is possible to avoid denominators in the above definition of f^{∗} if we replace the subgroup module 𝕊(G) by the Burnside ring Ω(G) of G. By definition, Ω(G) is the Grothendieck ring associated to the semi-ring Ω(G)^{+} consisting of isomorphism classes of G-sets, with addition and multiplication induced by the disjoint sum and direct product of G-sets, respectively; cf. Curtis-Reiner[CR2], ch. 11.

The additive structure of Ω(G) is easily understood, for Ω(G) is by [CR2], (80.6) a free abelian group with the canonical basis {[G/H]}_{H≤_{G}G}; in other words,

Ω(G) = ⨁_{H≤_{G}G} ℤ · [G/H], (7.7)

where the notation “H ≤_{G} G” means that the index set runs over a system of representatives of the set of conjugacy classes of all subgroups of G.

We have a natural surjection

s_{G} : 𝕊(G) → Ω(G)

which associates to each subgroup H ≤ G the isomorphism class of the G-set G/H; thus s_{G}(H) = [G/H]. It is clear that the map ch_{G} : 𝕊(G) → ch(ℚG) factors over s_{G}, so we have an induced map

ch_{Ω(G)} : Ω(G) → ch(ℚG),

whose kernel is denoted by

Ω(G)^{0} = Ker(ch_{Ω(G)});

thus, by definition we have

s_{G}^{−1}(Ω(G)^{0}) = 𝕊(G)^{0}.

More generally, we also have the ring homomorphism

π_{RG} : Ω(G) → Mod(RG)_{sym}

which associates to each G-set S its associated permutation module:

π_{RG}(S) = R[S]. (7.8)

From the definitions it is clear that we have

π_{RG} ∘ s_{G} = ch_{RG}. (7.9)

Remark 7.1 The ranks of the modules 𝕊(G)^{0} and Ω(G)^{0} are easily determined by using Artin’s Induction Theorem (cf. [KR1]):

(7.10) rank 𝕊(G)^{0} = #(subgroups of G)

−#(conjugacy classes of cyclic subgroups of G),

(7.11) rank Ω(G)^{0} = #(conjugacy classes of non-cyclic subgroups of G).

In particular, it follows that if G is cyclic, then 𝕊(G)^{0} = Ω(G)^{0} = 0 (and conversely).

If f : G_{1} → G_{2} is a group homomorphism, then we have an induced ring homomorphism

f^{∗} : Ω(G_{2}) → Ω(G_{1}),

which is obtained by restriction of operators: each G_{2}-set S is naturally a G_{1}-set via the rule g_{1} · s = f(g_{1})s.

To obtain a map in the other direction, let S be a G_{1}-set, and define the G_{2}-set G_{2} ×_{f} S as follows. Considering the cartesian product G_{2} × (Ker(f)\S) as a left f(G_{1})-set via the rule f(g_{1}) · (g_{2}, s) = (g_{2}f(g_{1})^{−1}, g_{1}s), we make the orbit space

G_{2} ×_{f} S = f(G_{1})\(G_{2} × (Ker(f)\S))

into a left G_{2}-set via g_{2} ·(f(G_{1})(g, s)) = f(G_{1})(g_{2}g, s). We thus obtain an additive map

f_{∗} : Ω(G_{1}) → Ω(G_{2})

by setting f_{∗}([S]) = [G_{2} ×_{f} S].

In the sequel it will be useful to know the matrix representations of f^{∗} and f_{∗} in terms of the canonical bases of Ω(G_{1}) and Ω(G_{2}). To this end we prove:

Proposition 7.2 Let f : G_{1} → G_{2} be a group homomorphism.

a) For each subgroup X ≤ G_{1} we have

f_{∗}([G_{1}/X]) = [G_{2}/f(X)]. (7.12)

b) For each subgroup Y ≤ G_{2} we have

f^{∗}([G_{2}/Y]) = ∑_{g∈f(G_{1})\G_{2}/Y} [G_{1}/f^{−1}(gY g^{−1})] = ∑_{X≤_{G_{1}} G_{1}} n_{f}(Y, X)[G_{1}/X], (7.13)

where n_{f}(Y, X) = #{g ∈ G_{2} : f^{−1}(gY g^{−1}) ∼ X}/([G_{1} : X] · |Y |).

Proof. a) Consider the map

t : G_{2} × (Ker(f)\G_{1}/X) → G_{2}/f(X)

defined by t(g_{2}, Ker(f)g_{1}X) = g_{2}f(g_{1})f(X). It is clear that t is constant on the f(G_{1})-orbits of the cartesian product and hence induces a map

t̄: G_{2} ×_{f} (G_{1}/X) → G_{2}/f(X).

Clearly, t̄is a G_{2}-equivariant surjection, and is easily checked that t̄is also injective. Thus G_{2} ×_{f} G_{1}/X ≃ G_{2}/f(X) as G_{2}-sets, and the assertion follows.

b) To prove the first equality, we shall use the decomposition

G_{2}/Y = ⋃_{g∈f(G_{1})\G_{2}/Y} f(G_{1})gY/Y

as G_{1}-sets. Now for each g ∈ G_{2}, the assignment t_{g}(g_{1}) = f(g_{1})gY induces a map

t_{g} : G_{1}/f^{−1}(gY g^{−1}) → f(G_{1})gY/Y,

which is clearly a surjection of G_{1}-sets. Moreover, t_{g} is injective because if t_{g}(g_{1}) = t_{g}(g′_{1}), then f(g′_{1})g = f(g_{1})gy with y ∈ Y, so g_{1}^{−1}g′_{1} ∈ f^{−1}(gY g^{−1}) and hence g′_{1}f^{−1}(gY g^{−1}) = g_{1}f^{−1}(gY g^{−1}). Thus t_{g} is an isomorphism of G_{1}-sets, which proves the first equality in (7.13).

By what has just been proved we have

∑_{g∈f(G_{1})\G_{2}/Y} [G_{1}/f^{−1}(gY g^{−1})] = ∑_{g∈G_{2}} \frac{1}{|f(G_{1})gY |}[G_{1}/f^{−1}(gY g^{−1})]

= ∑_{X≤_{G_{1}} G_{1}} ∑_{{g∈G_{2} \atop f^{−1}(gY g^{−1})∼X}} \frac{1}{|f(G_{1})gY |}[G_{1}/X].

Now if f^{−1}(gY g^{−1}) ∼ X, then |X| = |Ker(f)| · |f(G_{1}) ∩ gY g^{−1}| = |Ker(f)| · |Y | · |f(G_{1})|/|f(G_{1})gY | = |G_{1}| · |Y |/|f(G_{1})gY |, and so the coefficient of [G_{1}/X] is precisely n_{f}(Y, X), which proves the second equality of (7.13).

Corollary 7.3 If S_{i} is a G_{i}-set, then

(7.14)  π_{RG_{1}}(f^{∗}([S_{2}])) = Res_{f}(R[S_{2}]),
(7.15)  π_{RG_{2}}(f_{∗}([S_{1}])) = Coind_{f}(R[S_{2}]).

In particular, the maps f_{∗} and f^{∗} preserve character relations:

f_{∗}(Ω(G_{1})^{0}) ⊂ Ω(G_{2})^{0}; f^{∗}(Ω(G_{2})^{0}) ⊂ Ω(G_{1})^{0}. (7.16)

Proof. The first equation is clear from the definitions. To prove the second, we may assume S_{1} = G_{1}/X, where X ≤ G_{1}. Then by (7.12) and Corollary 4.15 we have

π_{RG_{2}}(f_{∗}([G_{1}/X])) = R[G_{2}/f(X)] = Coind_{f}(R[G_{1}/X]),

which proves (7.15). The last assertion follows from (7.14) and (7.15) by taking R = K and noting that π_{KG} = ch_{KG}.

Remark 7.4 It also follows from the proposition that the maps f_{∗} and f^{∗} on the Burnside rings Ω(G_{i}) agree with those defined on the subgroup modules 𝕊(G_{i}):

(7.17)  s_{G_{2}} ∘ f_{∗} = f_{∗} ∘ s_{G_{1}},
(7.18)  s_{G_{1}} ∘ f^{∗} = f^{∗} ∘ s_{G_{2}}.

Here the second equation is understood to hold only after tensoring all spaces and maps with ℚ.

7.3. The pairing δ on 𝕊(G)^{0} × Mod(RG)_{non−deg}. Recall that in the previous two sections we had constructed and studied the pairing

δ : Mod(RG)_{sym}^{0} × Mod(RG)_{non−deg} → Id(R, K) = K^{×}/R^{×}.

Since the homomorphism

ch_{RG} : 𝕊(G) → Mod(RG)_{sym}

maps 𝕊(G)^{0} into Mod(RG)_{sym}^{0} (cf. (7.3)), we can pull δ back to a pairing

δ̃ = δ̃_{RG} : 𝕊(G)^{0} × RG → Id_{R}(K)

via the rule δ̃(∑n_{H}H, M) = δ(ch_{RG}(∑n_{H}H), M). (7.19)

Whenever convenient, we shall denote this pairing just by δ = δ̃.

Since ch_{RG} = π_{RG} ∘ s_{G}, we see that the pairing factors over Ω(G)^{0} to yield a pairing which we shall also denote by δ:

δ : Ω(G)^{0} × RG → Id_{R}(K).

We first show that this pairing δ = δ̃ is precisely the pairing which was introduced in section 2.

Proposition 7.5 If (M, h) is a non-degenerate, L-valued hermitian RG-module, then for any character relation {n_{H}} ∈ 𝕊(G)^{0} we have

δ({n_{H}}, M) = ∏_{H≤G} disc(Inv_{H}(M, h))^{n_{H}} . (7.20)

Proof. By Example 4.4b) and the reciprocity formula (6.28) we have

δ({n_{H}}, M) = ∏_{H≤G} d_{RG}((R[G/H], h_{G/H}) ⊗ (M, h))^{n_{H}}
= ∏ d_{RG}(Ind_{H}^{G}(R, id) ⊗ (M, h))^{n_{H}}
= ∏ d_{RH}(Res_{H}^{G}(M, h))^{n_{H}} ,

which proves (7.20).

Next we shall use the maps f^{∗} and f_{∗} which were defined in the previous subsection to formulate the coinduction formulae. For this, recall the following notation from section 2:

Notation 7.6 For an element x = ∑ n_{X}X ∈ 𝕊(G)^{0} and a RG-module M, let

δ_{RG}^{∗}(x, M) = ε_{RG}(x, M)δ_{RG}(x, M), (7.21)

where the element ε_{RG}(x, M) ∈ K^{×} is defined by

ε_{RG}(x, M) = ∏_{X} |X|^{n_{X}rk(M^{X})}. (7.22)

Since ε is additive in x and is constant on conjugacy classes of subgroups, it follows that ε_{RG} and hence δ^{∗} factor over Ω(G). Explicitly we have for a G-set S:

ε_{RG}([S], M) = ∏_{s∈G\S} |G_{s}|^{rk(M^{Gs})}.

Note that if x is fixed, then ε_{RG}(x, M) depends only on the KG-module structure of M ⊗ K, and hence is additive on exact sequences.

Remark 7.7 If (M, h) is a non-degenerate hermitian RG-module, then it follows from Proposition 7.5 and Proposition 5.4d) that

δ_{RG}^{∗}({n_{H}}, M) = ∏_{H≤G} disc(M^{H}, h_{|M^{H}}))^{n_{H}} . (7.23)

Thus, δ^{∗} is the same invariant as that of Remark 2.6.

Theorem 7.8 (Coinduction) Let f : G_{1} → G_{2} be a group homomorphism, and let M_{i} be a non-degenerate RG_{i}-module. Then we have for any x_{i} ∈ Ω(G_{i})^{0} the formulae:

(7.24)  δ_{RG_{1}}(f^{∗}(x_{2}), M_{1}) = δ_{RG_{2}}(x_{2}, Coind_{f}(M_{1})),
(7.25)  δ_{RG_{2}}^{∗}(f_{∗}(x_{1}), M_{2}) = δ_{RG_{1}}^{∗}(x_{1}, Res_{f}(M_{2})).

In other words, we have

δ_{RG_{2}}(f_{∗}(x_{1}), M_{2}) = ε_{f}(x_{1}, M_{2})δ_{RG_{1}}(x_{1}, Res_{f}(M_{2})) (7.26)

where for x_{1} = ∑ n_{X}X ∈ 𝕊(G_{1}) the correction term ε_{f}(x_{1}, M_{2}) is given by

ε_{f}(x_{1}, M_{2}) = ε(x_{1}, Res_{f}(M_{2}))ε(f_{∗}(x_{1}), M_{2})^{−1} = ∏_{X≤G_{1}} |Ker(f) ∩ X|^{n_{X}rk(M_{2}^{f(X)})}.

(7.27)

Proof. In view of (7.14), the first equation follows directly from the Coinduction Theorem 6.15. To prove the second one, write x_{i} = [S_{1}] − [S_{2}]. Then by (7.15) we obtain from the Coinduction Theorem for Permutation Modules (Theorem 6.19) that

δ(f_{∗}(x_{1}), M_{2}) = δ_{RG_{2}}(Coind_{f}(R[S_{1}]), Coind_{f}(R[S_{2}]); M_{2})
= ε_{f}(S_{1}, M)ε_{f}(S_{2}, M)^{−1}δ_{RG_{1}}(R[S_{1}], R[S_{2}]; Res_{f}(M_{2})),

where, as in Theorem 6.19,

ε_{f}(S_{i}, M) = ∏_{s∈G_{1}\S_{i}} |Ker(f) ∩ (G_{1})_{s}|^{rk(H^{0}(f((G_{1})_{s}),M_{2}))}

= ∏_{s∈G_{1}\S_{i}} (\frac{|(G_{1})_{s}|}{|f((G_{1})_{s})|})^{rk(H^{0}(f((G_{1})_{s}),M_{2}))}
= ε([S_{i}], Res_{f}(M_{2}))ε(f_{∗}([S_{i}]), M_{2})^{−1};

here we have used (7.12) and the fact that H^{0}(f(X), M_{2}) = H^{0}(X, Res_{f}(M_{2})) for every X ≤ G_{1}. This proves (7.26) and (7.27), from which (7.25) follows readily.

7.4. Proofs of the main results. We can now proceed to prove the main results stated in section 2, with the exception of Theorem 2.7, which will be considered separately in the next section.

For the most part, the proofs just consist of translating the results of section 6 into the language of character relations, which is done as follows. Let ∑ n_{H}1_{H}^{∗} = 0 be a character relation, and put

M_{1} = ⨁_{n_{H}>0} Ind_{H}^{G}(R)^{n_{H}} , M_{2} = ⨁_{n_{H}<0} Ind_{H}^{G}(R)^{−n_{H}} .

Then M_{1} ⊗ K ≃ M_{2} ⊗ K and we have in the notation of sections 5 and 6:

δ({n_{H}}, M) = δ(M_{1}, M_{2}; M);

this follows immediately from the definitions (cf. (7.19) and (5.22)).

Proof of Theorem 2.5: Since by Proposition 7.5 we have

δ(M_{1}, M_{2}; M) = ∏_{H} disc_{R}(Inv_{H}(M, h))^{n_{H}} ,

the assertions of Theorem 2.5 follow immediately from those of Theorem 6.1.

Proof of Theorem 2.8: This follows immediately from the more precise Uniform Boundedness Theorem 6.6.

Proof of Theorem 2.9: This is a special case of the Triviality Theorem 6.23.

Proof of Theorem 2.10 and of Corollary 2.11: These follow from the Exact Sequence Formula (Theorem 6.21) and Corollary 6.31, combined with the fact that by Shapiro’s lemma (cf. [CE], ch. XII, Ex. 7) we have

H^{q}(G, M_{i} ⊗ M) = ⨁_{X≤G} H^{q}(X, Ind_{X}^{G}(M))^{⊕n_{X}} ≃ ⨁_{X≤G} H^{q}(X, M)^{⊕n_{X}} , for q ≥ 0.

Proof of Theorem 2.12: Let f : X → G denote the inclusion map. Then by the definition of f_{∗} in subsection 7.1 we have

ind_{X}^{G}(∑ m_{Y} Y ) = f_{∗}(∑ m_{Y} Y ),

and so (2.14) follows directly from (7.26) (and (7.17)). (Note that (7.27) shows that ε_{f}(x, M) = 1 if f is injective.) Moreover, by (7.6) we have

res_{X}^{G}(∑ n_{H}H) = ∑_{Y ≤X} ∑_{H≤G} n_{H}m(Y, H)Y = ∑_{H≤G} n_{H}f^{∗}(H) = f^{∗}(∑ n_{H}H),

so (2.15) follows from (7.24) and (7.18). This proves Theorem 2.12.

Proof of Example 2.13: a) This follows immediately from Theorem 2.12 by noting that if X is cyclic, then res_{X}^{G}({n_{H}}) ∈ Ω(X)^{0} = 0 by (7.16) and Remark 7.1.

b) The hypothesis means that M = R[S] is a direct sum of modules induced from cyclic subgroups (cf. Example 4.4b)), and so the assertion follows from a).

c) By decomposing S into its orbits, we may reduce to the case that G acts transitively on S. Then R[S] = Ind_{X}^{G}(R) by Example 4.4b), where X = G_{s}, and so we obtain by applying Theorem 2.12 with N = R that

δ_{R}({n_{H}}, R[S]) = δ({n_{Y}^{∗} }, R) = ∏_{Y} |Y |^{−n_{Y}^{∗}}

by (2.7). Now by (2.16) we have

∏_{Y} |Y |^{−n_{Y}^{∗}} = ∏_{H} ∏_{Y} |Y |^{−n_{H}m(H,Y)}
= ∏_{H} ∏_{g∈G} |gHg^{−1} ∩ X|^{−n_{H}/(|H|[X:gHg^{−1}∩X])}
= ∏_{H} ∏_{g∈H\G/X} |H ∩ gXg^{−1}|^{−n_{H}} ,

which proves (2.19).

Proof of Theorem 2.14: a) We shall apply Theorem 7.8 to the quotient map f : G → Q = G/X. By definition of f^{∗} (cf. (7.6)) we have

f^{∗}( ∑_{Y ≤Q} n_{Y} Y ) = ∑_{H≤G} ∑_{Y ≤Q} n_{Y} m_{f}(Y, H)H.

Now m_{f}(Y, H) = 0 unless H ∼ f^{−1}(Y), and in this case

m_{f}(Y, H) = #{g ∈ Q : f^{−1}(gY g^{−1}) = H}/([G : H]|Y |)
= #{g ∈ Q : gY g^{−1} = f(H)}/([G : H]|Y |)
= |N_{Q}(f(H))|/([G : H]|Y |)
= |N_{G}(H)|/|G| = [G : N_{G}(H)]^{−1}

since H ≥ X. Thus

f^{∗}( ∑_{Y ≤Q} n_{Y} Y ) = ∑_{X≤H≤G} ∑_{Y ∼f(X)} n_{Y} /[G : N_{G}(H)]H = inf_{G}^{Q}({n_{Y} }).

Since Coind_{f}(M) = M^{X} by Proposition 4.12b), the assertion (2.20) follows immediately from (7.24).

b) Here we have

f_{∗}(∑_{H} n_{H}H) = ∑_{H} n_{H}f(H) = ∑_{Y} ∑_{{H \atop f(H)=Y}} n_{H}Y = ind_{G}^{Q}(\{n_{H}\}),

so equations (2.21) – (2.23) follow directly from equations (7.25) – (7.27).

Proof of Example 2.15: a) Apply Theorem 2.14b) with X = G.

b) By hypothesis, M may be viewed as an RQ-module, so the assertion follows from (2.22) because ind_{G}^{Q}({n_{H}}) ∈ 𝕊(Q)^{0} = 0 by (7.16) and Remark 7.1.

Proof of Corollary 2.16: Here we have rank(M^{H}) = 0 for all subgroups H ≤ G, so ε_{X}({n_{H}}, M) = 1. This proves the first assertion, and the second follows from Example 2.15b).

Proof of Proposition 2.18: Let H_{0}, …, H_{p} denote the p+1 subgroups of G of order p. We then have the relation

∑_{i=0}^{p} N_{H_{i}} = N_{G} + p · 1,

where N_{H} = ∑_{g∈H} g ∈ RG. This yields ∑ n_{H}ε_{H} = 0, where ε_{H} = \frac{1}{|H|}N_{H} ∈ KG, and so we obtain ∑ n_{H}1_{H}^{∗} = 0 because for any irreducible character χ we have

(∑n_{H}1_{H}^{∗}, χ)_{G} = χ(∑n_{H}ε_{H}) = 0.

Thus x = ∑ n_{H}H ∈ 𝕊(G)^{0} is a character relation. Moreover, since rk(𝕊(G)^{0}) = 1 by Remark 7.1, it follows that every character relation of G is a rational and hence an integral multiple of x because some n_{H} = 1.

a) We first note that since M_{𝔭} = (M^{(p)})_{𝔭} for 𝔭 = (p), it follows from the localization formula (2.2) that δ(M) = δ(M^{(p)}), and so we may assume henceforth that M = M^{(p)} = ℤ/p^{r}ℤ. Let X = Ker(α : G → Aut(M)), where α denotes the map defining the action of G on M. If X ≠ 1, then G/X is cyclic and M^{X} = M, so δ(M) = ℤ by Corollary 2.16.

Thus, assume that X = 1. Here G is a subgroup of Aut(M) = (ℤ/p^{r}ℤ)^{×}, so Aut(M) cannot be cyclic, which forces the conditions p = 2 and r ≥ 3. Moreover, in that case we have that α induces an isomorphism α : G →^{∼} Aut(M)_{2} = {±id_{M}, ±(1 + 2^{r−1})id_{M}}, so if we choose σ, τ ∈ G such that α(σ) = −id, α(τ) = (1 + 2^{r−1})id, then we have that

M^{⟨σ⟩} = M^{⟨στ⟩} = M^{G} = M_{2} and M^{⟨τ⟩} = 2M. (7.28)

We thus obtain δ(M) = (|M^{⟨σ⟩}| · |M^{⟨τ⟩}| · |M^{⟨στ⟩}| · |M|^{−1} · |M^{G}|^{−2})^{−1}ℤ = (2 · 2^{r−1} · 2 · 2^{−r} · 2^{−2})^{−1}ℤ = 2ℤ, which proves a).

b) By the exact sequence formula (Theorem 2.10) we have

δ(M) = δ(M_{tor})δ(\overline{M})ψ(π_{M})^{2}, (7.29)

where ψ(π_{M}) = ∏_{H} χ(Coker(π_{M}^{H} : M^{H} → \overline{M}^{H}))^{n_{H}}. Since \overline{M}^{H} = \overline{M}, we have the exact sequence

0 → M_{tor}/M_{tor}^{H} → M/M^{H} → Coker(π_{M}^{H}) → 0,

from which we obtain the relation

χ(Coker(π_{M}^{H})) = χ(M/M^{H})χ(M_{tor}/M_{tor}^{H})^{−1} = χ(M/M^{H})χ(M_{tor})^{−1}χ(M_{tor}^{H})

and hence

ψ(π_{M}) = χ(M_{tor})^{−Σ n_{H}} ∏ χ(M_{tor}^{H})^{n_{H}} ∏ χ(M/M^{H})^{n_{H}}
= δ(M_{tor})^{−1} ∏ χ(M/M^{H})^{n_{H}} ,

where we have used the fact that ∑ n_{H} = 0. Substituting this in (7.29) yields

δ(M) = δ(M_{tor})^{−1}δ(R)^{r} ∏_{H} χ(M/M^{H})^{2n_{H}} ,

because \overline{M} = R^{r}, where r = rk(M). Moreover, by (2.7) we have

δ(R) = ∏_{H} |H|^{−n_{H}} R = (∏_{i=0}^{p} |H_{i}|^{−1}) |1|^{−(−1)}|G|^{−(−p)}R = p^{−(p+1)+2p}R

and so b) follows.

c) The non-degeneracy hypothesis means that V ≃ V^{∗}, where V = \overline{M} ⊗ K. Since V^{∗} ≃ V^{⊗(p−1)}, we see that this is only possible for p = 2. This proves the first assertion.

To prove the next one, put X = Ker(G → Gl(\overline{M})) = Ker(G → Gl(V)); by hypothesis we have that X ≠ G. Moreover, since dim(V) = 1, it follows by elementary representation theory that G/X is cyclic, so |X| = p = 2. This proves the second statement.

To prove the first equality in equation (2.29), we shall use the exact sequence formula (7.29) again. Here we have

ψ(π_{M}) = χ(Coker(M^{X} → \overline{M})) = χ(\overline{M}/\overline{M^{X}}) = χ(M/(M_{tor} + M^{X})),

because V^{H} = \overline{M}^{H} = 0 if H ≠ X, \{1\} and Coker(M → \overline{M}) = 0, and so by (7.29) we obtain δ(M) = δ(M_{tor})δ(\overline{M})χ(M/(M_{tor} + M^{X}))^{2}. (7.30)

Next we compute δ(\overline{M}) via the formula (7.20). In order to apply this, we shall view \overline{M} = R as a hermitian RG-module via the standard pairing h(x, y) = xy. Since \overline{M}^{H} = 0 for H ≠ X, 1 and \overline{M}^{X} = \overline{M} we see that formula (7.20) reduces to

δ(\overline{M}) = disc(\overline{M}^{X}, \frac{1}{2}h) · disc(\overline{M}, h)^{−1} = \frac{1}{2}R

Substituting this in (7.30) proves the first equality of (2.29).

To prove the last equation, consider the exact sequence

M^{X} → \overline{M} → H^{1}(X, M_{tor}) → H^{1}(X, M) → H^{1}(X, \overline{M}).

Since X acts trivially on \overline{M} we have H^{1}(X, M) = Hom(X, \overline{M}) = 0 because \overline{M} is R-torsionfree. We thus obtain the exact sequence

0 → \overline{M}/\overline{M^{X}} → H^{1}(X, M_{tor}) → H^{1}(X, M) → 0 (7.31)

which yields the relation

χ(\overline{M}/\overline{M^{X}}) = χ(H^{1}(X, M_{tor})) · χ(H^{1}(X, M))^{−1}. (7.32)

Now since X is a cyclic group, the theory of Herbrand quotients (cf. Serre[Se], VIII.4) is applicable, and so by Serre[Se], VIII.4, Proposition 8 and its corollary we have:

χ(H^{1}(X, M_{tor})) = χ(Ĥ^{0}(X, M_{tor})),
χ(Ĥ^{0}(X, M))χ(H^{1}(X, M))^{−1} = χ(Ĥ^{0}(X, \overline{M}))χ(H^{1}(X, \overline{M}))^{−1} = 2R.

Here the last equality is valid because \overline{M} = R is a trivial RX-module and so

Ĥ^{0}(X, \overline{M}) = R/2R and H^{1}(X, \overline{M}) = 0.

Substituting these relations in (7.32) yields

χ(\overline{M}/\overline{M^{X}}) = χ(Ĥ^{0}(X, M_{tor})) · 2χ(Ĥ^{0}(X, M))^{−1},

and so the second equality of (2.29) follows.

d) Put Y = Ker(G → Aut(M_{4})). Then Y ≠ G because X acts non-trivially on M_{4}, and Y ≠ 1 because Aut(M_{4}) = ℤ/2ℤ is cyclic. Thus Y = ⟨τ⟩ is cyclic of order 2, which proves the first assertion.

To prove formula (2.30), put d = [M : (M_{tor} + M^{X})]. We shall show that the formulae

(7.33)  d · |Ĥ^{0}(Y, M)| = |Ĥ^{0}(Y, M_{tor})|
(7.34)  δ(M_{tor}) · |Ĥ^{0}(Y, M_{tor})| = 2ℤ

are valid, from which (2.30) follows immediately because by (2.29) we have

δ(M_{tor})δ(M) = \frac{1}{2}[δ(M_{tor})d]^{2} = \frac{1}{2}[2|Ĥ^{0}(Y, M)|^{−1}]^{2}.

Moreover, it follows from (7.33) and (7.34) that |Ĥ^{0}(Y, M)| divides 2 because δ(M_{tor}) is integral by a), and so the last assertion of d) is also evident.

It thus suffices to prove formulae (7.33) and (7.34). For the first, it is clearly enough to construct the exact sequence

0 → M/(M_{tor} + M^{X}) →^{N̄_{Y}} Ĥ^{0}(Y, M_{tor}) → Ĥ^{0}(Y, M) → 0. (7.35)

To construct this sequence we shall start with the obvious sequence

0 → N_{Y} (M)/N_{Y} (M_{tor}) → Ĥ^{0}(Y, M_{tor}) → Ĥ^{0}(Y, M) → 0 (7.36)

which is exact because M^{Y} = M_{tor}^{Y}. Next we observe that

N_{Y} (M^{X}) ⊂ M^{Y} ∩ 2M_{tor} ⊂ N_{Y} (M_{tor}), (7.37)

and hence N_{Y} induces a surjective homomorphism

N̄_{Y} : M/(M_{tor} + M^{X}) → N_{Y} (M)/N_{Y} (M_{tor}),

which will be seen to be an isomorphism.

To prove (7.37), let us first verify the following fact:

(σ − 1)x + (1 + τ)x ∈ 2M_{tor}, ∀x ∈ M. (7.38)

To see this, let a be a generator of M_{tor} = ℤ/mℤ. Then σ(a) = −a and τ(a) = ka with (k, m) = 1; we note that k ≡ 1 mod 4 because by definition of τ we have τ(\frac{m}{4}a) = \frac{m}{4}a, so k \frac{m}{4} ≡ \frac{m}{4} mod m. Now since (σ − 1)x, (1 + τ)x ∈ M_{tor}, we can write (σ − 1)x = k_{1}a, (1 + τ)x = k_{2}a for some integers k_{1}, k_{2}. Then

\begin{aligned}στ(x) & = & σ(−x + k_{2}a) & = & −(x + k_{1}a) − k_{2}a, \\ τσ(x) & = & τ(x + k_{1}a) & = & −x + k_{2}a − kk_{2}a, \end{aligned}

which yields the relation (kk_{1} + k_{2})a = −(k_{1} + k_{2})a or k_{1}(1 + k) + 2k_{2} ≡ 0 mod m. Since k ≡ 1 mod 4 and 4|m this gives k_{1} + k_{2} ≡ 0 mod 2, which proves (7.38).

From (7.38) it follows immediately that N_{Y} (M^{X}) ⊂ M^{Y} ∩ 2M_{tor}, which is the first inclusion of (7.37). Next, suppose x ∈ M^{Y} ∩ 2M_{tor}. Then x = 2y with y ∈ M_{tor} and we have τ(y) − y = z ∈ M_{2}. Thus x = 2y = N_{Y} (y) − x ∈ N_{Y} (M_{tor}) because N_{Y} (M_{4}) = M_{2}, and so (7.37) follows.

We thus obtain the desired map N̄_{Y}. To prove that N̄_{Y} is injective, it is enough to show that Ker(N_{Y} : M → M) ⊂ M_{tor} + M^{X}. (7.39)

This again follows from (7.38). Indeed, if x ∈ Ker(N_{Y}), then by (7.38) we have (σ − 1)x = 2y with y ∈ M_{tor}, so σ(x + y) = x + 2y + σ(y) = x + y, which means that x + y ∈ M^{X}. Thus x = −y + (x + y) ∈ M_{tor} + M^{X}, which proves (7.39). This shows that N̄_{Y} is an isomorphism, and so we obtain the desired exact sequence (7.35).

It remains to prove equation (7.34), for which we shall distinguish two cases:

Case 1: G does not act faithfully on M^{(2)}. Since Ỹ := Ker(G → Aut(M^{(2)})) ⊂ Ker(G → Aut(M_{4})) = Y, and Ỹ ≠ 1 by hypothesis, it follows that Ỹ = Y. Thus Y acts trivially on M^{(2)} and hence

Ĥ^{0}(Y, M_{tor}) = Ĥ^{0}(Y, M^{(2)}) = M^{(2)}/2M^{(2)} ≃ ℤ/2ℤ.

Since we have δ(M_{tor}) = ℤ by a), this proves (7.34).

Case 2: G acts faithfully on M^{(2)}. Here we shall show that Ĥ^{0}(Y, M_{tor}) = 0, (7.40)

from which (7.34) follows immediately because by a) we have in this case that δ(M_{tor}) = 2ℤ.

To prove (7.40), first note that since τ acts trivially on M_{4}, it follows from (7.28) that (M^{(2)})^{Y} = 2M^{(2)}. Moreover, since a′ = \frac{m}{2^{r}} a is a generator of M^{(2)}, we see that N_{Y} (a′) = a′+ka′ = 2(\frac{1+k}{2} )a′ is a generator of 2M^{(2)} because k ≡ 1 mod 4, and so N_{Y} (M^{(2)}) = 2M^{(2)}. Thus Ĥ^{0}(Y, M_{tor}) = Ĥ^{0}(Y, M^{(2)}) = 0, as desired.

This concludes the proof of equation (7.34) and hence, as was explained above, also of Proposition 2.18.

8. Applications to class number relations

In this final section we apply the theory of discriminants of hermitian ℤG-modules to study relations among S-regulators of number fields. For this, we shall first introduce a hermitian structure on the group of S-units and relate its discriminant to the S-regulator. This will then be used to compute the δ-invariant of U_{S}(K), from which Theorem 2.7 (and hence Brauer’s Theorem 1.2) follows readily. Finally, we shall prove Dirichlet’s Theorem 1.1.

8.1. The S-unit group. Let K be a number field and let S ⊃ S_{∞} be a finite set of places of K containing the archimedean places. As usual, the S-unit group of K is defined by

U_{S}(K) = {x ∈ K : |x|_{v} = 1, ∀v ∉ S}.

Here we shall view it as a bilinear ℤ-module via the pairing ρ_{S} defined by

(8.1) ρ_{S}(u, u′) = ∑_{v∈S} d_{v} log |u|_{v} log |u′|_{v}, for u, u′ ∈ U_{S}(K),

where d_{v} = [K̂_{v} : ℚ_{v}] denotes the absolute degree of the completion K̂_{v} and | · |_{v} is the absolute value associated to v which is normalized in such a way that its restriction to ℚ coincides with the usual p-adic or archimedean absolute values of ℚ. Thus, with this normalization the product formula for u ∈ U_{S}(K) becomes:

∏_{v∈S} |u|_{v}^{d_{v}} = 1. (8.2)

We now want to compute the discriminant of the bilinear module (U_{S}(K), ρ_{S}). As one might expect, its essential ingredient is the S-regulator of K, which, as in Tate[Ta], p. 22, is defined by

reg_{S}(K) = | det((d_{v} log |u_{i}|_{v}){1≤i≤r \atop v∈S\v_{0}})|, (8.3)

where v_{0} ∈ S is an arbitrarily chosen place and u_{1}, …, u_{r} denotes a basis of \overline{U}_{S}(K) = U_{S}(K)/U_{S}(K)_{tor}; recall that by the S-unit theorem its rank is r = rk(U_{S}(K)) = |S| − 1.

Proposition 8.1 The discriminant of the bilinear ℤ-module (U_{S}(K), ρ_{S}) is given by the formula

disc(U_{S}(K), ρ_{S}) = reg_{S}(K)^{2}d(S)/(w(K)p(S))ℤ, (8.4)

where w(K) = |U_{S}(K)_{tor}| = |U_{S_{∞}}(K)_{tor}| denotes the number of roots of unity in the field K,

d(S) = ∑_{v∈S} d_{v} and p(S) = ∏_{v∈S} d_{v}.

Proof. Let u_{1}, …, u_{r} be a basis of \overline{U}_{S}(K) = U_{S}(K)/U_{S}(K)_{tor}, and let G = (g_{ij}) denote the associated Gram matrix; thus

g_{ij} = ∑_{v∈S} d_{v} log |u_{i}|_{v} log |u_{j}|_{v}, for 1 ≤ i, j ≤ r.

Then by (3.38) and (3.37) we have

disc(U_{S}(K), ρ_{S}) = |U_{S}(K)_{tor}|^{−1} det(G)ℤ = w(K)^{−1} det(G)ℤ. (8.5)

In order to relate det(G) to the regulator, write S = {v_{1}, …, v_{r+1}} and consider the matrix

A = ( \begin{matrix}d_{v_{1}} log |u_{1}|_{v_{1}} & ⋯ & d_{v_{1}} log |u_{r}|_{v_{1}} & d_{v_{1}} \\ ⋮ & & ⋮ & ⋮ \\ d_{v_{r+1}} log |u_{1}|_{v_{r+1}} & ⋯ & d_{v_{r+1}} log |u_{r}|_{v_{r+1}} & d_{v_{r+1}} \end{matrix} ) ,

which is related to G by the formula

A^{t}diag(\frac{1}{d_{v_{1}}} , …, \frac{1}{d_{v_{r+1}}} )A = ( \begin{matrix}G & 0 \\ 0 & d(S) \end{matrix} );

this follows by multiplying out the left hand side and applying (8.2). We thus obtain: det(G) = det(A)^{2}/(d(S)p(S)). (8.6)

Finally we want to compute det(A). By adding the first r rows to the last one and applying (8.2) again, we obtain

det(A) = det ( \begin{matrix}d_{v_{1}} log |u_{1}|_{v_{1}} & ⋯ & d_{v_{1}} log |u_{r}|_{v_{1}} & d_{v_{1}} \\ ⋮ & & ⋮ & ⋮ \\ d_{v_{r}} log |u_{1}|_{v_{r}} & ⋯ & d_{v_{r}} log |u_{r}|_{v_{r}} & d_{v_{r}} \\ 0 & ⋯ & 0 & d(S) \end{matrix} ) = ±reg_{S}(K) · d(S).

Substituting this in (8.6) and (8.5), the assertion follows.

From now on we shall consider the case that S is invariant under a group G ≤ Aut(K) of automorphisms of K, so U_{S}(K) is a naturally a ℤG-module. It is immediate that ρ_{S} is G-equivariant, and thus (U_{S}(K), ρ_{S}) is in fact a hermitian ℤG-module.

To compute its δ-invariant, we shall determine the invariant modules of Ũ_{S}(K) with respect to subgroups H ≤ G.

Proposition 8.2 For each subgroup H ≤ G there is a natural isometry

i_{H} : (U_{H\S}(K^{H}), ρ_{H\S}) →^{∼} Inv_{H}(U_{S}(K), ρ_{S}), (8.7)

where we view H\S as the set of places of the fixed field K^{H} which lie below S.

Proof. Clearly U_{S}(K)^{H} = U_{S}(K) ∩ K^{H} = U_{H\S}(K^{H}), and hence the inclusion map i_{H} : U_{H\S}(K^{H}) → U_{S}(K) induces an isomorphism

i_{H} : U_{H\S}(K^{H}) →^{∼} U_{S}(K)^{H}.

To compare the hermitian structures, let u, u′ ∈ US̄(K^{H}), where S̄ = H\S. Then

ρ_{S}(i_{H}(u), i_{H}(u′)) = ∑_{v̄∈S̄} ∑_{v|v̄} d_{v} log |u|v̄ log |u′|v̄ = |H|ρS̄(u, u′),

because

∑_{v|v̄} d_{v} = dv̄|H|. (8.8)

Thus inv_{H}(ρ_{S}) = ρS̄, which proves the assertion.

Corollary 8.3 Let x = ∑ n_{H}H ∈ 𝕊(G)^{0} be a character relation. Then

δ(x, U_{S}(K)) = ∏_{H} (\frac{d(H\S)reg_{H\S}(K^{H})^{2}}{p(H\S)w(K^{H})} )^{n_{H}} . (8.9)

Proof. By Proposition 7.5 and Proposition 8.2 we have

δ(x, U_{S}(K)) = ∏_{H} disc(Inv_{H}(U_{S}(K), ρ_{S}))^{n_{H}} = ∏_{H} disc(U_{H\S}(K^{H}), ρ_{H\S})^{n_{H}} ,

and so (8.9) follows from Proposition 8.1.

The above formula (8.9) can be simplified by means of the following lemma.

Lemma 8.4 Let S be a G-invariant set of places of K, and let x = ∑ n_{H}H ∈ 𝕊(G)^{0} be a character relation. Then:

(8.10)  ∏_{H} d(H\S)^{n_{H}} = δ(x, ℤ),
(8.11)  ∏_{H} p(H\S)^{n_{H}} = δ(x, ℤ[S]).

Proof. a) If H ≤ G is a subgroup, then by (8.8) we have

d(S) = |H|d(H\S),

and so we obtain

∏_{H} d(H\S)^{n_{H}} = ∏_{H}(|H|^{−1}d(S))^{n_{H}} = (∏_{H} |H|^{−n_{H}} ) · (∏_{H} d(S)^{n_{H}} ) = δ(x, ℤ);

here we have used (2.7) and the fact that

∑_{H} n_{H} = ∑_{H} n_{H}(1_{G}, 1_{H}^{∗})_{G} = 0. (8.12)

b) By decomposing S into its G-orbits and treating each orbit separately, we may assume without loss of generality that G acts transitively on S. Thus, if we fix v_{0} ∈ S and let Z = G_{v_{0}} denote the stabilizer (or decomposition group) of v_{0}, then the map g → gv_{0} induces a bijection G/Z →^{∼} S. Moreover, since the decomposition group of v = gv_{0} is gZg^{−1}, we obtain the relation

d_{v} = |gZg^{−1}|d_{w} = |Z|d_{w}, (8.13)

where w is the place of K^{G} lying below any v ∈ S.

If H ≤ G is any subgroup, then the H-orbit space S̄ = H\S may be identified with the double coset space H\G/Z, and for each v̄ ∈ S̄ we have the analogous formula d_{v} = |H ∩ gZg^{−1}|dv̄, if v = gv_{0}|v̄, (8.14)

because Z_{H}(v) = H ∩ gZg^{−1} is the decomposition group of v = gv_{0} with respect to H. Comparing (8.14) with (8.13) gives

dv̄ = \frac{|Z|d_{w}}{|H ∩ gZg^{−1}|}, (8.15)

and so we obtain

p(H\S) = ∏_{v̄∈H\S} dv̄ = ∏_{g∈H\G/Z} \frac{|Z|d_{w}}{|H ∩ gZg^{−1}|}. (8.16)

This yields

∏_{H} p(H\S)^{n_{H}} = ∏_{H} ∏_{g∈H\G/Z} (\frac{|Z|d_{w}}{|H ∩ gZg^{−1}|})^{n_{H}} = ∏_{H} ∏_{g∈H\G/Z} |H ∩ gZg^{−1}|^{−n_{H}} ,

where the latter equality follows from the fact that

∑_{H} n_{H}|H\G/Z| = ∑_{H} n_{H}(1_{H}^{∗}, 1_{Z}^{∗})_{G} = 0. (8.17)

(In the last formula we have used the fact that |H\G/Z| = (1_{H}^{∗}, 1_{Z}^{∗}) by Mackey’s Theorem and Frobenius Reciprocity.) In view of Example 2.13c), this proves (8.11).

8.2. Proofs of the main results. We can now proceed to prove the main theorem on class number relations:

Proof of Theorem 2.7: Write x = ∑ n_{H}H. Then by Corollary 8.3 and Lemma 8.4 we obtain

(8.18) δ(x, U_{S}(K)) = ∏_{H} (\frac{d(H\S)reg_{H\S}(K^{H})^{2}}{p(H\S)w(K^{H})} )^{n_{H}}

= ∏_{H} (\frac{reg_{H\S}(K^{H})^{2}}{w(K^{H})} )^{n_{H}} δ(x, ℤ)δ(x, ℤ[S])^{−1},

which proves the first assertion of the theorem. To deduce equation (2.8) from this, we first note that Brauer’s formula (1.6) may be generalized to S-regulators and S-class numbers in the following manner:

∏_{H} (reg_{H\S}(K^{H})h_{H\S}(K^{H})w(K^{H})^{−1})^{n_{H}} = 1. (8.19)

This follows from the usual Artin formalism applied to the Artin L-functions L_{S}(s, χ) (as defined in Tate[Ta], p. 23), using the fact that

L_{S}(s, 1_{H}^{∗}) = ζ_{K^{H},H\S}(s) ∼ −\frac{h_{H\S}(K^{H})reg_{H\S}}{w(K^{H})} s^{|H\S|−1}

in a neighbourhood of s = 0 (cf. [Ta], p. 23).

Thus, by (8.19) and (8.18) we obtain

∏_{H} h_{H\S}(K^{H})^{2n_{H}} = ∏_{H} w(K^{H})^{n_{H}} ∏_{H} (\frac{reg_{H\S}(K^{H})^{2}}{w(K^{H})} )^{−n_{H}}
= δ(x, U(K)_{tor})^{−1} (δ(x, U_{S}(K))δ(x, ℤ)^{−1}δ(x, ℤ[S]))^{−1} ,

which proves equation (2.8). The last assertion follows immediately from Example 2.13b).

As an illustration of the above class number formula, let us now prove Dirichlet’s Theorem:

Proof of Theorem 1.1: Since G = Gal(K/ℚ) ≃ ℤ/2ℤ × ℤ/2ℤ, we obtain from the first assertion of Proposition 2.18 and Theorem 2.7 the class number relation

[h(ℚ(\sqrt{d}))h(ℚ(\sqrt{−d}))h(ℚ(i))h(K)^{−1}h(ℚ)^{−2}]^{2} = 2[δ(U(K))δ(U(K)_{tor})]^{−1}, (8.20) where we have also used the fact that δ(ℤ) = 2ℤ; cf. Proposition 2.18b).

In order to compute the right hand side of 8.20, we shall apply Proposition 2.18d) to M = U(K), and thus need to check that M satisfies the hypotheses there. For this, note first that \overline{M} is as in Proposition 2.18c) because by Dirichlet’s Unit Theorem we have rk(U(K)) = r = r_{1} + r_{2} − 1 = 1 since r_{1} = 2 and r_{2} = 1, and hence G acts non-trivially on \overline{M} since rk(M^{G}) = rk(U(ℚ)) = 0. Moreover, if K^{+} = ℚ(\sqrt{d}) denotes the real subfield of the CM-field K, then rk(U(K^{+})) = 1, so the σ of 2.18c) is just the complex conjugation automorphism. Next we note that M_{tor} is cyclic, being generated by a suitable root of unity ζ of K, and so σ acts by multiplication by −1 since σ(ζ) = ζ̄ = ζ^{−1}. In addition we see that 4 divides |M_{tor}| because i = \sqrt{−1} ∈ K. Thus M satifies all the hypotheses of Proposition 2.18d), and so substituting (2.30) in the right hand side of (8.20) and using the well-known fact that h(ℚ(i)) = h(ℚ) = 1 yields the relation

[h(d)h(−d)h(K)^{−1}]^{2} = |Ĥ^{0}(Y, M)|^{2}.

We thus see that (1.1) holds with

Q = 2|Ĥ^{0}(Y, M)|^{−1}. (8.21)

It remains to verify that this definition of Q coincides with the one given in (1.2). For this, we first note that Y = Gal(K/ℚ(i)) because Y fixes M_{4} = ⟨i⟩. Thus, if K ≠ ℚ(ζ_{8}), then U(K)^{Y} = U(ℚ(i)) = ⟨i⟩ and N_{Y} (U(K)) = ⟨±1, N_{Y} (ε)⟩, and so the desired coincidence is immediate in view of the last assertion of Proposition 2.18d). On the other hand, if K = ℚ(ζ_{8}), then M_{tor} = M_{8} = ⟨ζ_{8}⟩, and so by the proof of Proposition 2.18d) (or otherwise) it follows that Ĥ^{0}(Y, M) = Ĥ^{0}(Y, M_{tor}) = 0. We thus obtain from (8.21) that Q = 2 in this case, which proves the assertion.

Remark 8.5 The above proof also shows that the invariant Q can be interpreted as an index of unit groups. Indeed, if K ≠ ℚ(ζ_{8}), then it follows from (2.30) and (2.29) that

Q = [U(K) : (U(K)_{tor} + U(K^{+}))],

which is the usual index considered in the theory of abelian fields and CM-fields (cf. Washington[Wash], p. 39).

In order to include also the case K = ℚ(ζ_{8}), we note with Herglotz[He] (cf. also Kubota[Kb]) that the above index is equal to the index

Q = [U(K) : Ũ(K)],

where Ũ(K) denotes the subgroup generated by the unit groups of the proper subfields of K, and that this formula is also true for K = ℚ(ζ_{8}).

We have thus also proven Herglotz’s version of Dirichlet’s Theorem. Note that this version is actually considerably easier to obtain by our methods than Dirichlet’s version because it uses essentially only Proposition 2.18c) and not part d) which required a much more detailed analysis.

Discriminants of hermitian R[G]-modules 87

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