THE RING OF WITT VECTORS

Leslie G. Roberts

I discuss the ring of (big or universal) Witt vectors Λ(A), of a commutative ring A. The ring Λ(A), or variants of it, arises in several ways in Commutative Algebra, Algebraic Number Theory, Algebraic Geometry, Algebraic Topology, and Algebraic K-Theory, and undoubtedly elsewhere. I have been fascinated by Witt Vectors for many years, but they are still very mysterious objects to me. I will not to attempt a more detailed motivation here, but some of the applications hinted at above appear below, and in the references.

These notes are based on talks I gave in the Curves Seminar in January, February, and March of 1996. For some time I have wanted to give at least an informal exposition of the material in Section 5. I was motivated to give the talks at this time by the appearance of divided powers in the lectures of Tony Geramita [5,Lecture 9]. Nothing here is claimed to be original except for the speculation in Section 5 on the relation between the P ’s and the Q’s. However there is some novelty in the presentation. In particular I have not seen the universal polynomials Q defined as they are here in (4.2), (4.3.3). I learned about this approach from Barry Dayton, but have not seen it in print. Several accounts of Witt vectors are listed in the bibliography. Much of these notes are my attempt to understand Bergman’s sections on Witt vectors in [1]. The denominator f^{n}g^{n} in (4.3.1), (4.4) seems to be a satisfactory answer to Bergman’s lament in the footnote at the bottom of page 190 in [1]. Grothendieck also has this correction term in [2, page 29 (in the numbering at the bottom of the page), (1.17 bis)].

I would like to thank the audience for attending my lectures and (especially Ernst Kani) for offering helpful suggestions. My talks got to the end of Section 8. Section 9 was added later in order to reach a more natural stopping point.

Throughout, ℕ = {1, 2, 3, …}, the natural numbers.

1. The additive structure of Λ(A).

Let A be a commutative ring with unit. A formal power series over A is an expression of the form f = a_{0} + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … = Σ_{i=0}^{∞}a_{i}t^{i} (a_{i} ∈ A). If also g = b_{0} + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … we define f + g by just adding the coefficients, i.e.

(1.1)  f + g := Σ_{i=0}^{∞}(a_{i} + b_{i})t^{i}.

We also define the product by

(1.2)  fg := Σ_{i=0}^{∞}c_{k}t^{k} where c_{k} = Σ_{i+j=k}a_{i}b_{j}.

Note that the expression for c_{k} is a finite sum so our definition of the product makes sense. We do not assume that A has a topology (and if it does, we ignore the topology). These operations make the set of formal power series over A into a commutative ring with unit which is denoted A[[t]]. A polynomial f = a_{0} + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} is regarded as an element of A[[t]] by taking a_{i} = 0 for i >n.

Let Λ(A) = 1 + tA[[t]] be the set of all formal power series with constant term 1. An element f of Λ(A) is a (possibly) infinite sum f = 1+a_{1}t+a_{2}t^{2} +a_{3}t^{3} +…. First observe that Λ(A) is an abelian group under multiplication of formal power series. The “zero” element for the abelian group Λ(A) is the power series 1. We obtain the “negative” of f ∈ Λ(A) by formally inverting power series: if f = 1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} + … we wish to find g = 1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … such that fg = 1. Setting the coefficients of t^{i} (in fg) equal to zero we obtain equations

(1.3)  a_{1} + b_{1} = 0
a_{2} + a_{1}b_{1} + b_{2} = 0
a_{3} + a_{2}b_{1} + a_{1}b_{2} + b_{3} = 0
….

These equations can be solved recursively for the b_{i} yielding

(1.4)  b_{1} = −a_{1}
b_{2} = a_{1}^{2} − a_{2}
b_{3} = −a_{1}^{3} + 2a_{1}a_{2} − a_{3}
….

I do not know a useful explicit formula for the b_{j} as functions of the a_{i}. Note that b_{j} depends only on {a_{1}, …, a_{j}} and that the coefficients of these polynomials are integers which are independent of A.

2. The ghost map.

We now try to change this multiplicative abelian group into an additive abelian group. One way to do this is to take logarithms. If ℚ ⊆ A we can define log(f) by substituting f −1 in place of x in the formal power series log(1+x) := x− \frac{x^{2}}{2} + \frac{x^{3}}{3} − \frac{x^{4}}{4} +. . .. This substitution yields a series log(f) ∈ tA[[t]] because only the finite sum x − \frac{x^{2}}{2} + \frac{x^{3}}{3} − \frac{x^{4}}{4} + … + (−1)^{i} \frac{x^{i}}{i} contributes to the coefficient of t^{i}. Similarly, again assuming that ℚ ⊆ A, for g ∈ tA[[t]] we can define e^{g} ∈ Λ(A) by substituting g in place of x in the series e^{x} := 1+x+ \frac{x^{2}}{2!} + \frac{x^{3}}{3!} +…. For convenience of typing I will sometimes write exp(g) instead of e^{g}. The usual properties of exp and log are satisfied. In particular we have (for f_{i} ∈ Λ(A), g_{i} ∈ tA[[t]])

(2.1.1)  log(f_{1}f_{2}) = log(f_{1}) + log(f_{2})
(2.1.2)  exp(g_{1} + g_{2}) = exp(g_{1}) exp(g_{2})
(2.1.3)  exp(log(f_{1})) = f_{1}
(2.1.4)  log(exp(g_{1})) = g_{1}

Furthermore, given f = a_{0} + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … ∈ A[[t]] we can define f′ (= \frac{df}{dt}) by f′ := a_{1} + 2a_{2}t + 3a_{3}t^{2} + …. There is no limit involved here. This is purely an algebraic definition. Then all the usual properties of differentiation are satisfied, including

(2.1.5)  \frac{d}{dt} log(f) = \frac{f′}{f}

If we use log we require that ℚ ⊆ A. However because \frac{d}{dt}(g_{1} + g_{2}) = \frac{d}{dt}(g_{1}) + \frac{d}{dt}(g_{2}), we have that f → \frac{d}{dt} log(f) is also a homomorphism from the multiplicative group Λ(A) to the additive group A[[t]]. This homomorphism does not require that ℚ ⊆ A, since \frac{f}{f} ′ is defined over any commutative ring. Furthermore, from the rules of differentiation we have fg → \frac{(fg)′}{fg} = \frac{f′g+fg′}{fg} = \frac{f′}{f} + \frac{g′}{g} which checks that f → \frac{f′}{f} is a homomorphism without using property (2.1.1) of log. Thus for any commutative ring A and f ∈ Λ(A) we define the homomorphism

(2.2)  gh : Λ(A) → tA[[t]]

by gh(f) = −\frac{tf}{f} ′. Here Λ(A) is an abelian group under multiplication of formal power series, as described above, and tA[[t]] is an abelian group under addition of formal power series. The factor (−t) does not change the fact that we have a homomorphism, and will turn out to be convenient later. Because of the t, we land in tA[[t]] instead of A[[t]]. The name “gh” stands for “ghost”. Suppose that f = 1 + Σ_{i=1}^{∞}a_{i}t^{i}, and gh(f) = Σ_{i=1}^{∞}w_{i}t^{i}. From the formula f · gh(f) = −tf′ we obtain

(1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + …)(w_{1}t + w_{2}t^{2} + w_{3}t^{3} + …) = −a_{1}t − 2a_{2}t^{2} − 3a_{3}t^{3} + …

Multiplying out and comparing coefficients of powers of t we obtain equations

(2.3)  w_{1} + a_{1} = 0
w_{2} + a_{1}w_{1} + 2a_{2} = 0
. . . . . . .
w_{n} + a_{1}w_{n−1} + … a_{n−1}w_{1} + na_{n} = 0

If we are given {a_{i}}_{i≥1} we can solve these equations recursively for {w_{j}}_{j≥1}, obtaining

(2.3.1)  w_{i} = W_{i}(a_{1}, …, a_{i})

where W_{i} is a polynomial in i variables, with coefficients in ℤ. For example w_{1} = −a_{1}, w_{2} = a_{1}^{2} − 2a_{2}, …. Furthermore W_{i} is independent of A. Similarly if ℚ ⊆ A we can recursively solve (2.3) for the a_{i}, obtaining

(2.3.2)  a_{i} = A_{i}(w_{1}, …, w_{i})

where the A_{i} are polynomials independent of A with coefficients in ℚ.

Presumably gh is called the ghost map because the co-ordinates w_{i} (“ghost coordinates”) of f ∈ Λ(A) do not really exist unless ℚ ⊆A. Nonetheless we will see that one can sometimes prove something about Λ(A) by working with the w_{i} as if they did exist.

If ℚ ⊆ A then gh is an isomorphism. We can find the inverse by solving the equation g = −\frac{tf′}{f} for f. Namely \frac{−g}{t} = \frac{f′}{f} = \frac{d}{dt} log(f), − ∫ \frac{g}{t} = log(f) and f = exp(− ∫ \frac{g}{t}). Note that \frac{g}{t} ∈ A[[t]] and by ∫ applied to an element of A[[t]] we mean the antiderivative with constant of integration 0, i.e. ∫ (b_{0} + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + …) = b_{0}t + b_{1} \frac{t^{2}}{2} + b_{2} \frac{t^{3}}{3} + b_{3} \frac{t^{4}}{4} + . . .. If g ∈ tA[[t]] we will write gh^{−1}(g) for exp(− ∫ \frac{g}{t}). It is straightforward to check that gh^{−1} is a two sided inverse to gh so gh is an isomorphism of abelian groups:

(2.4)  gh : Λ(A) ⟶^{≅} tA[[t]] (if ℚ ⊆ A)

3. The ring structure on Λ(A).

The abelian group tA[[t]] can be made into a ring in several different ways. First of all we can identify tA[[t]] with Π_{i=1}^{∞}A by

a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … ⇔ (a_{1}, a_{2}, a_{3}, …)..

The set Π_{i=1}^{∞}A is a ring under co-ordinatewise addition and multiplication. In this ring structure on Π_{i=1}^{∞}A the variable t is just a place holder. (Of course t is still needed in the definitions of gh and gh^{−1}.) If g_{1} and g_{2} are elements of tA[[t]] I will denote this co-ordinatewise product by g_{1} · g_{2}. Then (still assuming that ℚ ⊆ A) we can define a multiplication * on Λ(A) by

(3.1)  f_{1} ∗ f_{2} = gh^{−1}(gh(f_{1}) · gh(f_{2})).

It is then automatic that Λ(A) is a (commutative) ring (with unit), addition being the usual multiplication of power series, and multiplication *. We will now try to describe f ∗g more explicitly. Suppose f = 1+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+…+a_{n}t^{n}+… and g = 1+b_{1}t+b_{2}t^{2}+b_{3}t^{3}+… Then f ∗ g = 1 + c_{1}t + c_{2}t^{2} + c_{3}t^{3} + … + c_{n}t^{n} + …. From (2.3.*) it is clear that there exist polynomials P_{n} ∈ ℚ[X_{1}, …, X_{n}; Y_{1}, …, Y_{n}], independent of A, such that

(3.2)  c_{n} = P_{n}(a_{1}, …, a_{n}; b_{1}, …, b_{n}).

Thus * is determined by its behaviour on polynomials. In order to find c_{n} it suffices to compute f ∗ g where f = 1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} and g = 1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … + b_{n}t^{n}. To get started we will consider (1 + a_{1}t) ∗ (1 + b_{1}t). We have

gh(1 + a_{1}t) = \frac{(−t)(1 + a_{1}t)}{(1 + a_{1}t)}′ = (−a_{1}t)(1 − a_{1}t + a_{1}^{2}t^{2} − a_{1}^{3}t^{3} + …) =

−a_{1}t + a_{1}^{2}t^{2} − a_{1}^{3}t^{3} + ….

The negative signs are a bit of a nuisance so instead we consider

gh(1 − a_{1}t) = a_{1}t + a_{1}^{2}t^{2} + a_{1}^{3}t^{3} + …. Similarly
gh(1 − b_{1}t) = b_{1}t + b_{1}^{2}t^{2} + b_{1}^{3}t^{3} + …. Hence

(1 − a_{1}t) ∗ (1 − b_{1}t) = gh^{−1}(a_{1}b_{1}t + a_{1}^{2}b_{1}^{2}t^{2} + a_{1}^{3}b_{1}^{3}t^{3} + …) = 1 − a_{1}b_{1}t. That is

(3.3)  (1 − a_{1}t) ∗ (1 − b_{1}t) = 1 − a_{1}b_{1}t.

The unit element of tA[[t]] is clearly t + t^{2} + t^{3} + … and gh(1 − t) = t + t^{2} + t^{3} + … so 1 − t is the unit element of the ring Λ(A).

Now take f = 1+a_{1}t+a_{2}t^{2} +a_{3}t^{3} +…+a_{n}t^{n} and g = 1+b_{1}t+b_{2}t^{2} +b_{3}t^{3} +…+b_{n}t^{n}. Since (3.2) is independent of the ring we can take the a_{i}, b_{j} to be indeterminates over A. More precisely, let {λ_{i}, μ_{i}} (1 ≤ i ≤ n,) be independent indeterminates over A, and let a_{i} = (−1)^{i}(i^{th} elementary symmetric function in the λ’s), b_{i} = (−1)^{i}(i^{th} elementary symmetric function in the μ’s). For example, if n = 3 then a_{1} = −λ_{1} − λ_{2} − λ_{3}, a_{2} = λ_{1}λ_{2}+λ_{1}λ_{3}+λ_{2}λ_{3} and a_{3} = −λ_{1}λ_{2}λ_{3}. Then the a_{i}, b_{i} are independent indeterminates over A and f = Π_{i=1}^{n}(1 − λ_{i}t) and g = Π_{i=1}^{n}(1 − μ_{i}t). (The λ_{i} can be thought of as (reciprocal) formal roots of f.) Now using the distributive law for the ring Λ(A[{λ_{i}}, {μ_{j}}]) we obtain that f ∗ g = Π_{1≤i,j≤n}(1 − λ_{i}t) ∗ (1 − μ_{j}t) = Π_{1≤i,j≤n}(1 − λ_{i}μ_{j}t) = 1 + c_{1}t + c_{2}t^{2} + c_{3}t^{3} + … + c_{n}t^{n} + c′_{n+1}t^{n} + … + c′_{n^{2}}t^{n^{2}} . (The c_{i} are correct as required in (3.1) only up to t^{n}, hence the change in notation to c′_{i} if i > n. If we require c_{k} for k > n just take n ≥k.) We now have that the c_{i} (1 ≤ i ≤ n) are symmetric functions in {λ_{i}} and in {μ_{i}} (1 ≤ i ≤ n), and hence are polynomials in {a_{i}, b_{i}}(1 ≤ i ≤ n). That is, the polynomials P_{i} in (3.2) have integer coefficients. The first few examples are

(3.4)  P_{1}(a_{1}; b_{1}) = −a_{1}b_{1}
P_{2}(a_{1}, a_{2}; b_{1}, b_{2}) = a_{1}^{2}b_{2} − 2a_{2}b_{2} + a_{2}b_{1}^{2}
P_{3}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}, b_{3}) = −(a_{1}^{3}b_{3} + a_{3}b_{1}^{3} − 3a_{1}a_{2}b_{3} − 3a_{3}b_{1}b_{2} + 3a_{3}b_{3} + a_{1}a_{2}b_{1}b_{2})

….

The P_{n} rapidly become very complicated (see (3.5) below for a more extensive list), and as far as I know there is no closed form expression for them. However it is clear from the construction that P_{n} is homogeneous of degree n in the a’s if we assign degree i to a_{i} and also homogeneous of degree n in the b’s if we assign degree i to b_{i}. Also P_{n} is invariant under interchanging a_{i} and b_{i} (1 ≤ i ≤ n).

Note that gh(Π_{i=1}^{n}(1 − λ_{i}t)) = Σ_{i=1}^{n}gh(1 − λ_{i}t) = Σ_{i=1}^{n}Σ_{j=1}^{∞}λ_{i}^{j}t^{j} = Σ_{j=1}^{∞}(Σ_{i=1}^{n}λ_{i}^{j})t^{j} from which we see that w_{j} = Σ_{i=1}^{n}λ_{i}^{j}. Thus the w_{j} is the j^{th} power sum of the λ_{i} and a_{j} is (up to sign) the j^{th} elementary symmetric function of the λ_{i}. Indeed the formulas (2.3) are just Newton’s formulas relating these symmetric functions (taking into account the signs).

The polynomials P_{n} determine the ring structure of Λ(A). If f = 1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} + … and g = 1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … + b_{n}t^{n} + … then

f ∗ g = 1 + Σ_{i≥1}P_{i}(a_{1}, …, a_{i}; b_{1}, …, b_{i})t^{i}

(here the a’s are no longer indeterminates over A, but specific elements of A which we plug into the polynomials). Since the P_{i} have coefficients in ℤ we can use this as the definition of f ∗ g even if ℚ is not contained in A. I claim that ∗ makes Λ(A) into a ring. The ring axioms can all be expressed as identities in the P_{n}. For example consider the associative law for multiplication:

[(1 + a_{1}t) ∗ (1 + b_{1}t)] ∗ (1 + c_{1}t) = (1 + P_{1}(a_{1}; b_{1})t) ∗ (1 + c_{1}t) = 1 + P_{1}(P_{1}(a_{1}; b_{1}); c_{1})t.

Similarly (1+a_{1}t)∗[(1+b_{1}t)∗(1+c_{1}t)] = (1+a_{1}t)∗(1+P_{1}(b_{1}; c_{1})t) = 1+P_{1}(a_{1}; P_{1}(b_{1}; c_{1}))t. The associative law for multiplication in Λ(A) is thus equivalent to P_{1}(P_{1}(a_{1}; b_{1}); c_{1}) = P_{1}(a_{1}; P_{1}(b_{1}; c_{1})) for all rings A and all a_{1}, b_{1}, c_{1} ∈ A. These identities are known to hold over ℚ for all a_{1}, b_{1}, c_{1} ∈ ℚ, and hence we have an equality P_{1}(P_{1}(X; Y); Z) = P_{1}(X; P_{1}(Y; Z)) in ℚ[X, Y, Z] for indeterminates X, Y, Z. The coefficients of P_{1} actually lie in ℤ so we have P_{1}(P_{1}(X; Y); Z) = P_{1}(X; P_{1}(Y; Z)) in ℤ[X, Y, Z]. Hence we have P_{1}(P_{1}(a_{1}; b_{1}); c_{1}) = P_{1}(a_{1}; P_{1}(b_{1}; c_{1})) for all elements a_{1}, b_{1}, c_{1} in any commutative ring, as we wished to show. (Of course P_{1}(X; Y) = XY so the above identities are all trivial. My intention is only to illustrate the idea.) There are similar identities involving P_{n} for n ≥ 1, from which we conclude that for any commutative ring A, Λ(A) is a commutative ring with unit, the addition of Λ(A) being multiplication of formal power series, and the multiplication being *. We still have a map

gh : Λ(A) → tA[[t]]

of commutative rings, which we claim is a ring homomorphism. We already know that gh(f_{1} + f_{2}) = gh(f_{1}) + gh(f_{2}) so it suffices to show that gh(f_{1} ∗ f_{2}) = gh(f_{1}) · gh(f_{2}) for polynomials f_{1} and f_{2} of degree n in Λ(A). Suppose f_{1} = 1+Σ_{i=1}^{n}a_{i}t^{i} and f_{2} = 1+Σ_{i=1}^{n}b_{i}t^{i}. If ℚ ⊆ A we have gh(f_{1} ∗ f_{2}) = gh(f_{1}) · gh(f_{2}) by definition of *. This equality can be expressed as identities in the P ’s which hold for all a_{i}, b_{i} ∈ ℚ, hence which hold universally, proving that gh preserves multiplication as well as addition, hence is a ring homomorphism.

The homomorphism gh need not be either injective or surjective. For example, if the characteristic of A is p then gh(1 + t^{p}) = 0, and t^{2} is not in the image of gh unless 2 is a unit, as we see by setting w_{2} = 1, w_{1} = 0 in the second equation of (2.3).

The formula (3.3) determines the multiplication of Λ(A). Formula (3.3) could have been taken as true by definition, and the multiplication developed from there. However I find (3.1) easier to remember, and I can better understand proofs based on (3.1). Instead of (3.3) some expositions start with (1 + a_{1}t) ∗ (1 + a_{2}t) = (1 + a_{1}a_{2}t) [2], [3 (exercises)], [4]. This gives an isomorphic ring Λ(A) (via t → −t), but will necessitate assorted sign changes throughout the discussion.

(3.5) Examples. Here is a more extensive list of the P_{i}, calculated with Mathematica using (3.1) and (3.2). Here a[i] means a_{i}, b[i] means b_{i} and p[i] is P_{i}. The list is given far enough for the calculation in (5.3).

p[1] = -(a[1]*b[1])
p[2] = a[2]*b[1]^2 + a[1]^2*b[2] - 2*a[2]*b[2]
p[3] = -(a[3]*b[1]^3) - a[1]*a[2]*b[1]*b[2] + 3*a[3]*b[1]*b[2] -
    a[1]^3*b[3] + 3*a[1]*a[2]*b[3] - 3*a[3]*b[3]
p[4] = a[4]*b[1]^4 + a[1]*a[3]*b[1]^2*b[2] - 4*a[4]*b[1]^2*b[2] +
    a[2]^2*b[2]^2 - 2*a[1]*a[3]*b[2]^2 + 2*a[4]*b[2]^2 +
    a[1]^2*a[2]*b[1]*b[3] - 2*a[2]^2*b[1]*b[3] - a[1]*a[3]*b[1]*b[3] +

4*a[4]*b[1]*b[3] + a[1]^4*b[4] - 4*a[1]^2*a[2]*b[4] +
2*a[2]^2*b[4] + 4*a[1]*a[3]*b[4] - 4*a[4]*b[4]
p[5] = -(a[5]*b[1]^5) - a[1]*a[4]*b[1]^3*b[2] + 5*a[5]*b[1]^3*b[2] -
a[2]*a[3]*b[1]*b[2]^2 + 3*a[1]*a[4]*b[1]*b[2]^2 -
5*a[5]*b[1]*b[2]^2 - a[1]^2*a[3]*b[1]^2*b[3] +
2*a[2]*a[3]*b[1]^2*b[3] + a[1]*a[4]*b[1]^2*b[3] -
5*a[5]*b[1]^2*b[3] - a[1]*a[2]^2*b[2]*b[3] +
2*a[1]^2*a[3]*b[2]*b[3] + a[2]*a[3]*b[2]*b[3] -
5*a[1]*a[4]*b[2]*b[3] + 5*a[5]*b[2]*b[3] - a[1]^3*a[2]*b[1]*b[4] +
3*a[1]*a[2]^2*b[1]*b[4] + a[1]^2*a[3]*b[1]*b[4] -
5*a[2]*a[3]*b[1]*b[4] - a[1]*a[4]*b[1]*b[4] + 5*a[5]*b[1]*b[4] -
a[1]^5*b[5] + 5*a[1]^3*a[2]*b[5] - 5*a[1]*a[2]^2*b[5] -
5*a[1]^2*a[3]*b[5] + 5*a[2]*a[3]*b[5] + 5*a[1]*a[4]*b[5] -
5*a[5]*b[5]
p[6] = a[6]*b[1]^6 + a[1]*a[5]*b[1]^4*b[2] - 6*a[6]*b[1]^4*b[2] +
a[2]*a[4]*b[1]^2*b[2]^2 - 4*a[1]*a[5]*b[1]^2*b[2]^2 +
9*a[6]*b[1]^2*b[2]^2 + a[3]^2*b[2]^3 -
2*a[2]*a[4]*b[2]^3 + 2*a[1]*a[5]*b[2]^3 - 2*a[6]*b[2]^3 +
a[1]^2*a[4]*b[1]^3*b[3] - 2*a[2]*a[4]*b[1]^3*b[3] -
a[1]*a[5]*b[1]^3*b[3] + 6*a[6]*b[1]^3*b[3] +
a[1]*a[2]*a[3]*b[1]*b[2]*b[3] - 3*a[3]^2*b[1]*b[2]*b[3] -
3*a[1]^2*a[4]*b[1]*b[2]*b[3] + 4*a[2]*a[4]*b[1]*b[2]*b[3] +
7*a[1]*a[5]*b[1]*b[2]*b[3] - 12*a[6]*b[1]*b[2]*b[3] +
a[2]^3*b[3]^2 - 3*a[1]*a[2]*a[3]*b[3]^2 +
3*a[3]^2*b[3]^2 + 3*a[1]^2*a[4]*b[3]^2 -
3*a[2]*a[4]*b[3]^2 - 3*a[1]*a[5]*b[3]^2 +
3*a[6]*b[3]^2 + a[1]^3*a[3]*b[1]^2*b[4] -
3*a[1]*a[2]*a[3]*b[1]^2*b[4] + 3*a[3]^2*b[1]^2*b[4] -
a[1]^2*a[4]*b[1]^2*b[4] + 2*a[2]*a[4]*b[1]^2*b[4] +
a[1]*a[5]*b[1]^2*b[4] - 6*a[6]*b[1]^2*b[4] +
a[1]^2*a[2]^2*b[2]*b[4] - 2*a[2]^3*b[2]*b[4] -

    2*a[1]^3*a[3]*b[2]*b[4] + 4*a[1]*a[2]*a[3]*b[2]*b[4] -
    3*a[3]^2*b[2]*b[4] + 2*a[1]^2*a[4]*b[2]*b[4] +
    2*a[2]*a[4]*b[2]*b[4] - 6*a[1]*a[5]*b[2]*b[4] + 6*a[6]*b[2]*b[4] +
    a[1]^4*a[2]*b[1]*b[5] - 4*a[1]^2*a[2]^2*b[1]*b[5] +
    2*a[2]^3*b[1]*b[5] - a[1]^3*a[3]*b[1]*b[5] +
    7*a[1]*a[2]*a[3]*b[1]*b[5] - 3*a[3]^2*b[1]*b[5] +
    a[1]^2*a[4]*b[1]*b[5] - 6*a[2]*a[4]*b[1]*b[5] -
    a[1]*a[5]*b[1]*b[5] + 6*a[6]*b[1]*b[5] + a[1]^6*b[6] -
    6*a[1]^4*a[2]*b[6] + 9*a[1]^2*a[2]^2*b[6] - 2*a[2]^3*b[6] +
    6*a[1]^3*a[3]*b[6] - 12*a[1]*a[2]*a[3]*b[6] + 3*a[3]^2*b[6] -
    6*a[1]^2*a[4]*b[6] + 6*a[2]*a[4]*b[6] + 6*a[1]*a[5]*b[6] - 6*a[6]*b[6]

4. Another ring structure on Λ(A).

If ℚ ⊆ A we have the isomorphism

gh : Λ(A) ⟶^{≅} tA[[t]]

The right hand side tA[[t]] has other multiplications besides the co-ordinatewise multiplication · considered above. For example we might consider the usual power series multiplication of tA[[t]] (indicated simply by juxtaposition), which together with the addition yields a ring without unit. We can use this to make Λ(A) into a ring without unit, with addition being multiplication of power series, and multiplication ■ defined by

(4.1)  f_{1}■f_{2} = gh^{−1}(gh(f_{1})gh(f_{2}))

For example we have gh(1−t) = \frac{t}{1−t} . Squaring we get \frac{t^{2}}{(1−t)^{2}} . Dividing by t and integrating we get ∫ \frac{t}{(1−t)^{2}} = \frac{t}{1−t} + log(1 − t) (making sure that we take the antiderivative with 0 constant term). Taking exp(−this expression) we obtain (1 − t)■(1 − t) = exp(\frac{−t}{1−t})\frac{1}{1−t} = 1 − \frac{t^{2}}{2} + . . .. Thus we see that ■ requires denominators, and if f_{1} and f_{2} are polynomials f_{1}■f_{2} need not even be rational.

We have better luck if we use the multiplication ⊗ defined on tA[[t]] by t^{m} ⊗ t^{n} = \binom{m+n}{n}t^{m+n} . (The n^{th} power of t under ⊗ is then (n!)t^{n} and t^{n} is a divided power.) Use this to define a multiplication

(4.2)  f_{1}□f_{2} = gh^{−1}(gh(f_{1}) ⊗ gh(f_{2}))

on Λ(A). Then we have

gh(1 − (λ + μ)t) = Σ_{n=1}^{∞}(λ + μ)^{n}t^{n}
gh(1 − λt) = Σ_{n=1}^{∞}λ^{n}t^{n}
gh(1 − μt) = Σ_{n=1}^{∞}μ^{n}t^{n}

so that

gh\frac{(1 − (λ + μ)t)}{(1 − λt)(1 − μt)} = Σ_{n=1}^{∞}[(λ + μ)^{n} − λ^{n} − μ^{n}]t^{n} = Σ_{d≥2}(Σ_{i+j=d,i≥1,j≥1}\binom{d}{i}λ^{i}μ^{j})t^{d}

= (Σ_{n=1}^{∞}λ^{n}t^{n}) ⊗ (Σ_{n=1}^{∞}μ^{n}t^{n}).

Thus we see that

(4.3.1)  (1 − λt)□(1 − μt) = \frac{(1 − (λ + μ)t)}{(1 − λt)(1 − μt)}.

Replacing λ and μ by −λ, −μ we obtain

(4.3.2)  (1 + λt)□(1 + μt) = \frac{(1 + (λ + μ)t)}{(1 + λt)(1 + μt)}.

Replacing t by −t has the same effect, so we do not have the same sign convention problem we had in Section 3 with *. Again using (2.3) and the definition of ⊗ we find that if

f_{1} = 1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} + … and f_{2} = 1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … then

(4.3.3)  f_{1}□f_{2} = 1 + Σ_{n=1}^{∞}Q_{n}(a_{1}, …, a_{n−1}; b_{1}, …, b_{n−1})t^{n}

where the Q_{i} are polynomials independent of A with coefficients in ℚ. The polynomial Q_{n} cannot contain a_{n} or b_{n} since the coefficient of the t^{n} term of (Σ_{i=1}^{∞}u_{i}t^{i}) ⊗ (Σ_{i=1}^{∞}v_{i}t^{i}) does not contain u_{n} or v_{n}. We can thus compute Q_{n+1} (as well as Q_{i} for i < n + 1) by taking f_{1} and f_{2} to be polynomials of degree n. By temporarily taking indeterminates λ_{i}, μ_{i} over A and setting a_{i} = (−1)^{i}(i^{th} elementary symmetric function in the λ_{i}), b_{i} = (−1)^{i}(i^{th} elementary symmetric function in the μ_{i}) and using the distributive law in Λ(A) we conclude from (4.3.1) that (4.4) f_{1}□f_{2} = (1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n})□(1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … + b_{n}t^{n}) = ((1 − λ_{1}t)(1 − λ_{2}t) … (1 − λ_{n}t))□((1 − μ_{1}t)(1 − μ_{2}t) … (1 − μ_{n}t)) =

Π_{1≤i,j≤n}(1 − λ_{i}t)□(1 − μ_{j}t) = \frac{Π_{1≤i,j≤n}(1 − (λ_{i} + μ_{j})t}{f^{n}g^{n}}

The last expression is symmetric in the λ ’s and the μ’s from which we conclude that the coefficients of the Q’s in fact lie in ℤ. A heuristic reason for the denominator is that without it f□g would change when extra roots λ_{i} equal to zero are added.

Now let A be an arbitrary commutative ring. Since the Q’s have coefficients in ℤ, given two elements f_{1} = 1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … + a_{n}t^{n} + … and f_{2} = 1 + b_{1}t + b_{2}t^{2} + b_{3}t^{3} + … in Λ(A) we can define

(4.5)  f_{1}□f_{2} = 1 + Σ_{n=1}^{∞}Q_{n}(a_{1}, …, a_{n−1}; b_{1}, …, b_{n−1})t^{n}

The commutative and associative laws for □ , as well as the distributive law, and the fact that gh is a homomorphism of rings without unit are all expressed in terms of identities over ℤ amongst the Q’s. These identities are independent of A and are equalities of polynomials since they hold if A contains ℚ, which is infinite. Thus the same properties hold for any commutative ring. That is, for all commutative rings A, the operations of multiplication of formal power series (as addition) and □ (as multiplication) make Λ(A) into a commutative ring without unit, and gh is a ring homomorphism.

From the definition of □ it follows that P_{n} is homogeneous of degree n in the a’s and b’s combined if we assign degree i to a_{i} and to b_{i}. Also Q_{n} is invariant under interchanging a_{i} and b_{i} (1 ≤ i ≤ n). (This is the commutative law for □ ). Furthermore every non-zero term in Q_{n} has both a’s and b’s in it.

It is worth noting that the behaviour of the numerator in (4.4) is related to the way that Chern classes of vector bundles behave under tensor product. If V and W are two vector bundles of ranks m and n respectively with Chern polynomials f = 1 + c_{1}t + c_{2}t^{2} + c_{3}t^{3} + … + c_{m}t^{m} and g = 1 + d_{1}t + d_{2}t^{2} + d_{3}t^{3} + … + d_{n}t^{n} then the Chern polynomial of V ⊗ W is Π_{1≤i≤n,1≤j≤n}(1 + (λ_{i} + μ_{j})t) where f formally factors as f = Π_{i=1}^{n}(1 + λ_{i}t) and g as g = Π_{i=1}^{n}(1 + μ_{i}t). Similarly the Chern polynomial of V ⊕ W is fg (the usual multiplication of polynomials). Formal manipulations with Chern polynomials can then be used in the Schubert calculus to answer enumerative problems in Algebraic Geometry, for which the Schubert Maple package of Sheldon Katz may be useful.

I will not attempt to say what vector bundles are, or where V, W, c_{i}, d_{i}, λ_{i} or μ_{i} live. These remarks are intended only to give an idea of one of the motivations for formal considerations like those with Λ(A).

(4.6) Examples Here is a list of examples of the Q’s. These were calculated with Mathematica using (4.2) and (4.5). (I also did the calculation using the Schubert package, as described above, obtaining the same result). As in (3.5) q[i] means Q_{i}, a[i] is a_{i} and b[i] is b_{i}. This is the file newq.m1 loaded in the calculation of (5.3).

q[1] = 0
q[2] = -(a[1]*b[1])
q[3] = a[1]^2*b[1] - 2*a[2]*b[1] + a[1]*b[1]^2 - 2*a[1]*b[2]
q[4] = -(a[1]^3*b[1]) + 3*a[1]*a[2]*b[1] - 3*a[3]*b[1] -
a[1]^2*b[1]^2 + 3*a[2]*b[1]^2 - a[1]*b[1]^3 +
3*a[1]^2*b[2] - 6*a[2]*b[2] + 3*a[1]*b[1]*b[2] - 3*a[1]*b[3]
q[5] = a[1]^4*b[1] - 4*a[1]^2*a[2]*b[1] + 2*a[2]^2*b[1] +
4*a[1]*a[3]*b[1] - 4*a[4]*b[1] + a[1]^3*b[1]^2 -
4*a[1]*a[2]*b[1]^2 + 6*a[3]*b[1]^2 + a[1]^2*b[1]^3 -
4*a[2]*b[1]^3 + a[1]*b[1]^4 - 4*a[1]^3*b[2] +
12*a[1]*a[2]*b[2] - 12*a[3]*b[2] - 4*a[1]^2*b[1]*b[2] +
12*a[2]*b[1]*b[2] - 4*a[1]*b[1]^2*b[2] + 2*a[1]*b[2]^2 +
6*a[1]^2*b[3] - 12*a[2]*b[3] + 4*a[1]*b[1]*b[3] - 4*a[1]*b[4]
q[6] = -(a[1]^5*b[1]) + 5*a[1]^3*a[2]*b[1] - 5*a[1]*a[2]^2*b[1] -
5*a[1]^2*a[3]*b[1] + 5*a[2]*a[3]*b[1] + 5*a[1]*a[4]*b[1] -
5*a[5]*b[1] - a[1]^4*b[1]^2 + 5*a[1]^2*a[2]*b[1]^2 -
3*a[2]^2*b[1]^2 - 7*a[1]*a[3]*b[1]^2 + 10*a[4]*b[1]^2 -
a[1]^3*b[1]^3 + 5*a[1]*a[2]*b[1]^3 - 10*a[3]*b[1]^3 -
a[1]^2*b[1]^4 + 5*a[2]*b[1]^4 - a[1]*b[1]^5 +
5*a[1]^4*b[2] - 20*a[1]^2*a[2]*b[2] + 10*a[2]^2*b[2] +

    20*a[1]*a[3]*b[2] - 20*a[4]*b[2] + 5*a[1]^3*b[1]*b[2] -
    20*a[1]*a[2]*b[1]*b[2] + 30*a[3]*b[1]*b[2] + 5*a[1]^2*b[1]^2*b[2] -
    20*a[2]*b[1]^2*b[2] + 5*a[1]*b[1]^3*b[2] - 3*a[1]^2*b[2]^2 +
    10*a[2]*b[2]^2 - 5*a[1]*b[1]*b[2]^2 - 10*a[1]^3*b[3] +
    30*a[1]*a[2]*b[3] - 30*a[3]*b[3] - 7*a[1]^2*b[1]*b[3] +
    20*a[2]*b[1]*b[3] - 5*a[1]*b[1]^2*b[3] + 5*a[1]*b[2]*b[3] +
    10*a[1]^2*b[4] - 20*a[2]*b[4] + 5*a[1]*b[1]*b[4] - 5*a[1]*b[5]

5. The relation between the P’s and the Q’s.

Now I would like to consider the question of whether or not there is any relation between the P ’s and the Q’s. A few years ago Mark Krusemeyer, while visiting Queen’s, noticed the following:

(5.1.1)  P_{1} = Q_{2} = −a_{1}b_{1}

(5.1.2)  P_{2} = a_{2}b_{1}^{2} + a_{1}^{2}b_{2} − 2a_{2}b_{2} and Q_{3} = a_{1}^{2}b_{1} + a_{1}b_{1}^{2} − 2a_{2}b_{1} − 2a_{1}b_{2}.

We can change P_{2} to Q_{3} by the following rule: in a given monomial in the a’s and b’s lower the indices by one in all possible ways, with a_{1} → 0 and b_{1} → 0. Thus a_{2}b_{1}^{2} → a_{1}b_{1}^{2}, a_{1}^{2}b_{2} → a_{1}^{2}b_{1}, and −2a_{2}b_{2} → −2a_{2}b_{1} − 2a_{1}b_{2} so altogether P_{2} → Q_{3}.

(5.1.3)  P_{3} = −a_{1}^{3}b_{3} − a_{3}b_{1}^{3} + 3a_{1}a_{2}b_{3} + 3a_{3}b_{1}b_{2} − 3a_{3}b_{3} − a_{1}b_{1}a_{2}b_{2} and

Q_{4} = −6a_{2}b_{2} − a_{1}^{3}b_{1} − a_{1}b_{1}^{3} + 3a_{1}a_{2}b_{1} + 3a_{1}b_{1}b_{2} − a_{1}^{2}b_{1}^{2} − 3a_{3}b_{1} − 3a_{1}b_{3} + 3a_{1}^{2}b_{2} + 3a_{2}b_{1}^{2}.

This time we lower indices in a monomial by two in all possible ways, either one subscript by two or two subscripts by one each, again with a_{1} → 0 and b_{1} → 0. This time we get P_{3} → (−a_{1}^{3}b_{1})+(−a_{1}b_{1}^{3})+(3a_{1}^{2}b_{2}+3a_{1}a_{2}b_{1})+(3a_{1}b_{1}b_{2}+3a_{2}b_{1}^{2})+(−3a_{2}b_{2}−3a_{1}b_{3}−3a_{3}b_{1})+(−a_{1}^{2}b_{1}^{2}) which we see is equal to Q_{4} except for a term −3a_{2}b_{2} instead of −6a_{2}b_{2}. This is clearly too close to be a coincidence. One might continue, lowering subscripts by i-1 in an attempt to convert P_{i} to Q_{i+1}. We first remark that the degrees come out correctly: P_{i} is of degree 2i and 2i − (i − 1) = i + 1 = degree of Q_{i+1}. I tested more examples with the following Mathematica program, which gives a precise definition of the rule. (This is the file drop3.m loaded in the session log given below.) (5.2) Mathematica program.

drop[n , (x )[i ]] := x[i - n]
drop[0, M ] := M
drop[n , M Plus] := drop[n, First[M]] + drop[n, Rest[M]]
drop[n , (A Integer)*(M )] := A*drop[n, M]
drop[n , M ] := Module[{first, rest}, {first, rest} = decomp[M];
    Sum[drop[i, first]*drop[n - i, rest], {i, 0, n}]]
decomp[M ] := Module[{vars = Variables[M], first, rest},
    first = First[vars]; rest = M/first; {first, rest}]
a[i ] := 0 /; i < 1
b[i ] := 0 /; i < 1

Some examples are given below. Lines 4-9 verify the calculation changing P_{2} to Q_{3} (twice). Line 10 verifies the computation attempting to change P_{3} to Q_{4}. Line 11 applies the rule to P_{4} → Q_{5}. Note that the difference between Q_{5} and the rule applied to P_{4} is six terms out of 22. Line 13 applies the rule to P_{5} → Q_{6}. This time the error is 20 terms out of 43. (5.3) Mathematica session log Mathematica 2.2 for DOS 387

Copyright 1988-93 Wolfram Research, Inc.
In[1]:= << "math9\\drop3.m"
In[2]:= << "math11\\newp.m";
In[3]:= << "math11\\newq.m1";
In[4]:= drop[1,p[2]]

Out[4]= a[1]^{2} b[1] + a[1] b[1]^{2} - 2 (a[2] b[1] + a[1] b[2])
In[5]:= q[3]
Out[5]= a[1]^{2} b[1] - 2 a[2] b[1] + a[1] b[1]^{2} - 2 a[1] b[2]
In[6]:= %-%%
Out[6]= -2 a[2] b[1] - 2 a[1] b[2] + 2 (a[2] b[1] + a[1] b[2])
In[7]:= Expand[%]
Out[7]= 0
In[8]:= drop[1,p[2]]-q[3]
Out[8]= 2 a[2] b[1] + 2 a[1] b[2] - 2 (a[2] b[1] + a[1] b[2])
In[9]:= drop[1,p[2]]-q[3]//Expand
Out[9]= 0
In[10]:= drop[2,p[3]]-q[4]//Expand
Out[10]= 3 a[2] b[2]
In[11]:= drop[3,p[4]]-q[5]//Expand
Out[11]= -(a[1] a[2] b[1]^{2} ) - 4 a[1] a[2] b[2] + 8 a[3] b[2] -
    a[1]^{2} b[1] b[2] - 4 a[2] b[1] b[2] + 8 a[2] b[3]
In[12]:= q[5]
Out[12]= a[1]^{4} b[1] - 4 a[1]^{2} a[2] b[1] + 2 a[2]^{2} b[1] +
    4 a[1] a[3] b[1] - 4 a[4] b[1] + a[1]^{3} b[1]^{2} - 4 a[1] a[2] b[1]^{2} +
    6 a[3] b[1]^{2} + a[1]^{2} b[1]^{3} - 4 a[2] b[1]^{3} + a[1] b[1]^{4} -
    4 a[1]^{3} b[2] + 12 a[1] a[2] b[2] - 12 a[3] b[2] -
    4 a[1]^{2} b[1] b[2] + 12 a[2] b[1] b[2] - 4 a[1] b[1]^{2} b[2] +
    2 a[1] b[2]^{2} + 6 a[1]^{2} b[3] - 12 a[2] b[3] + 4 a[1] b[1] b[3] -
    4 a[1] b[4]
In[13]:= drop[4,p[5]]-q[6]//Expand

Out[13]= 2 a[1]^{2} a[2] b[1]^{2} - a[2]^{2} b[1]^{2} -
    3 a[1] a[3] b[1]^{2} + a[1] a[2] b[1]^{3} + 5 a[1]^{2} a[2] b[2] -
    5 a[2]^{2} b[2] - 10 a[1] a[3] b[2] + 15 a[4] b[2] +
    a[1]^{3} b[1] b[2] + 3 a[1] a[2] b[1] b[2] - 15 a[3] b[1] b[2] +
    2 a[1]^{2} b[1]^{2} b[2] + 5 a[2] b[1]^{2} b[2] -
    a[1]^{2} b[2]^{2} - 5 a[2] b[2]^{2} - 15 a[1] a[2] b[3] +
    25 a[3] b[3] - 3 a[1]^{2} b[1] b[3] - 10 a[2] b[1] b[3] +
    15 a[2] b[4]
In[14]:= Length[%]
Out[14]= 20
In[15]:= {Length[p[5]],Length[q[6]]}
Out[15]= {28, 43}

It seems clear that the rule defined above gives a first approximation to Q_{i+1}. However I have not been able to derive an exact rule, nor have I been able to explain the partial success of the present rule. I have tried some variants, but always with worse results. It seems unlikely that a rule (such as the one above) which has difference operators on subscripts (without taking into account the values of the subscripts) can be exact, since it seems (from the P_{3} → Q_{4} case) that a_{1}b_{1}a_{2}b_{2} is converted to a_{1}^{2}b_{1}^{2} whereas a_{3}b_{3} is converted to 2a_{2}b_{2} (plus other terms) yet both have a drop of one in each of two different subscripts.

6. Another set of co-ordinates for Λ(A).

Now consider the equation

(6.1)  1 + a_{1}t + a_{2}t^{2} + a_{3}t^{3} + … = (1 − x_{1}t)(1 − x_{2}t^{2})(1 − x_{3}t^{3}) …

If we compare coefficients of t^{n} (n ≥ 1) we obtain equations

(6.2)  a_{i} = Σ(−1)^{j}x_{i_{1}}x_{i_{2}} … x_{i_{j}} = R_{i}(x_{1}, …, x_{i}) (i ≥ 1)

where the summation is over all partitions i = i_{1} + i_{2} + … + i_{j} where i_{1} < i_{2} < … < i_{j}. For example

(6.3)  a_{1} = −x_{1}

a_{2} = −x_{2}
a_{3} = x_{1}x_{2} − x_{3}
. . . . . .

Since the equations have the form a_{i} = −x_{i}+ (terms involving {x_{j}}_{j<i}), we can solve the equations recursively for x_{i} (i ≥ 1) in any commutative ring A, obtaining x_{i} = S_{i}(a_{1}, …, a_{i}) where the S_{i} are polynomials with integer coefficients, independent of A. I do not know an explicit formula for the S_{i}. Thus the vector (x_{1}, x_{2}, x_{3}, ….) can be thought of as an alternate representation for f = 1+a_{1}t+a_{2}t^{2} +a_{3}t^{3} +… ∈ Λ(A). Using the polynomials R_{i} and S_{i} the operations of Λ(A) can be converted into operations on the vectors (x_{1}, x_{2}, x_{3}, ….) making the set Λ_{ℕ}(A) of such sequences into a commutative ring isomorphic to Λ(A). (See (6.11) for the notation Λ_{ℕ}.) Unless indicated otherwise I will continue to use juxtaposition to indicate the addition of this ring, and * to indicate the multiplication. The zero element (unit for addition) is (0, 0, 0, 0, …) and the 1 (unit for multiplication) is (1, 0, 0, 0, …). We will obtain that if \underline{x} = (x_{1}, x_{2}, x_{3}, ….) and \underline{y} = (y_{1}, y_{2}, y_{3}, ….) then \underline{xy} = (z_{1}, z_{2}, z_{3} …) and \underline{x} ∗ \underline{y} = (z′_{1}, z′_{2}, z′_{3} …) where z_{i} = z_{i}(x_{1}, …, x_{i}; y_{1}, …, y_{i}) and z′_{i} = z′_{i}(x_{1}, …, x_{i}; y_{1}, …, y_{i}) for universal polynomials with integer coefficients. I will spare the reader a new set of names! These polynomials can also be found by using the ghost map. We have

(6.4)  gh(1 − x_{i}t^{i}) = \frac{−ti(−x_{i}t^{i−1})}{1 − x_{i}t^{i}} = \frac{ix_{i}t^{i}}{1 − x_{i}t^{i}} = i(x_{i}t^{i} + x_{i}^{2}t^{2i} + x_{i}^{3}t^{3i} + …)

Then if gh(Π_{i≥1}(1 − x_{i}t^{i})) = w_{1}t + w_{2}t^{2} + w_{3}t^{3} + … we obtain by comparing coefficients of t^{i} with (6.4)

(6.5)  w_{n} = Σ_{i|n}ix_{i}^{n/i}

For example

(6.6)  w_{1} = x_{1}
w_{2} = x_{1}^{2} + 2x_{2}

w_{3} = x_{1}^{3} + 3x_{3}
w_{4} = x_{1}^{4} + 2x_{2}^{2} + 4x_{4}
….

These equations can be solved over ℚ for x_{n} in terms of w_{j} (j | n). If we then use gh and gh^{−1} to compute the above mentioned polynomials z_{i} = z_{i}(x_{1}, …, x_{i}; y_{1}, …, y_{i}) and z′_{i} = z′_{i}(x_{1}, …, x_{i}; y_{1}, …, y_{i}) we conclude that z_{i} and z′_{i} depend only on x_{j} and y_{j} for j | i. This makes these new polynomials simpler than the P ’s that we considered earlier, especially when i is a prime power. A few examples are

(6.7)  z_{1} = x_{1} + y_{1}
z_{2} = x_{2} − x_{1}y_{1} + y_{2}
z_{3} = x_{3} − x_{1}^{2}y_{1} − x_{1}y_{1}^{2} + y_{3}
(6.8)  z′_{1} = x_{1}y_{1}
z′_{2} = x_{2}y_{1}^{2} + x_{1}^{2}y_{2} + 2x_{2}y_{2}
z′_{3} = x_{3}y_{1}^{3} + x_{1}^{3}y_{3} + 3x_{3}y_{3}

Finally we should not forget that there are also universal polynomials for \underline{x}^{−1} (the inverse of \underline{x} in the abelian group Λ_{ℕ}(A)). Namely if as above \underline{x} = (x_{1}, x_{2}, x_{3}, ….) and \underline{x}^{−1} = (x′_{1}, x′_{2}, x′_{3}, ….) then x′_{i} is a polynomial with integer coefficients in x_{j} (j | i). A few examples are

(6.9)  x′_{1} = −x_{1}
x′_{2} = −x_{2} − x_{1}^{2}
x′_{3} = −x_{3}

The formulas (6.7), (6.8), (6.9) are copied from the attached computer printouts (6.12)-(6.14). From (1 − x_{1}t)(1 − x_{p}t^{p})(1 + x_{1}t)(1 + x_{p}t^{p}) = (1 − x_{1}^{2}t^{2})(1 − x_{p}^{2}t^{2p}) we conclude that x′_{p} = −x_{p} for all odd primes p, as suggested by the calculations in (6.9).

It is clear, from lifting to Λ(A) and multiplying, that if for all i either x_{i} = 0 or y_{i} = 0 then (x_{1}, x_{2}, …) + (y_{1}, y_{2}, …) = (x_{1} + y_{1}, x_{2} + y_{2}, ...). In particular (x_{1}, x_{2}, x_{3}, …) = (x_{1}, 0, 0, …) + (0, x_{2}, 0, …) + (0, 0, x_{3}, 0, …). (Here it is more convenient to use + for the addition of Λ(A).) From this it follows that in the expressions for z_{n} in (6.7), (6.12) every term except x_{n} and y_{n} contains both x_{i} and y_{i} for some i, 0 < i < n.

Let m, n ∈ ℕ (= {1, 2, 3, …}). From (6.4) we have gh(1 − x_{m}t^{m}) = m(x_{m}t^{m} + x_{m}^{2}t^{2m} + x_{m}^{3}t^{3m} + …) and gh(1 − x_{n}t^{n}) = n(x_{n}t^{n} + x_{n}^{2}t^{2n} + x_{n}^{3}t^{3n} + …) . When we multiply gh(1 − x_{m}t^{m}) and gh(1 − x_{n}t^{n}) coefficientwise the only non-zero terms will be in co-ordinates that are multiples of N=LCM(m, n). Thus gh(1 − x_{m}t^{m}) · gh(1 − x_{n}t^{n}) = mnΣ_{k≥1}x_{m}^{Nk/m}x_{n}^{Nk/n}t^{Nk} = gh((1 − x_{m}^{N/m}x_{n}^{N/n}t^{N})^{mn/N}). From this we conclude by the usual universal argument that

(6.10) (1 − x_{m}t^{m}) ∗ (1 − x_{n}t^{n}) = (1 − x_{m}^{N/m}x_{n}^{N/n}t^{N})^{mn/N} = (1 − x_{m}^{n/M}x_{n}^{m/M}t^{N})^{M} (where M=GCD(m, n)).

An example of (6.10) is (1 − at^{4}) ∗ (1 − bt^{6}) = (1 − a^{3}b^{2}t^{12})^{2}. Using (6.10), it is easy to show that z′_{p} = x_{1}^{p}y_{p} + x_{p}y_{1}^{p} + px_{p}y_{p} for all primes p, as suggested by the calculations in (6.8).

(6.11) Truncation. Let S be a non-empty subset of ℕ = {1, 2, 3, …} such that if n ∈ S then any divisor of n (in ℕ) is also in S. Some examples of such sets S are ℕ itself, \underline{P} = \{1, p, p^{2}, ….\} = \{p^{i}\}_{i≥0}, \underline{P}_{n} = \{1, p, p^{2}, …, p^{n}\}. Necessarily 1 ∈ S. Since the universal polynomials (6.7), (6.8), (6.9) depend only on the divisors of the integer in question we can use these polynomials to put a ring structure on Λ_{S}(A) := {(x_{1}, x_{s_{2}}, x_{s_{3}}, …)} where S = {1, s_{2}, s_{3}, …}. There is a surjective ring homomorphism π_{S} : Λ(A) → Λ_{S}(A) given by π_{S}(Π_{i≥1}(1 − x_{i}t^{i})) = (x_{1}, x_{s_{2}}, x_{s_{3}}, …). When computing in Λ_{S}(A) it is sometimes helpful to lift to Λ(A), using (6.10) to calculate *.

(6.12) The formulas (6.7) by computer.

X[1] = x[1] + y[1]
X[2] = x[2] - x[1]*y[1] + y[2]
X[3] = x[3] - x[1]^2*y[1] - x[1]*y[1]^2 + y[3]
X[4] = x[4] - x[1]^3*y[1] + x[1]*x[2]*y[1] - 2*x[1]^2*y[1]^2 -

x[1]*y[1]^3 - x[2]*y[2] + x[1]*y[1]*y[2] + y[4]
X[5] = x[5] - x[1]^4*y[1] - 2*x[1]^3*y[1]^2 -
2*x[1]^2*y[1]^3 - x[1]*y[1]^4 + y[5]
X[6] = x[6] - x[1]^5*y[1] + x[1]*x[2]^2*y[1] +
x[1]^2*x[3]*y[1] - 3*x[1]^4*y[1]^2 - x[1]^2*x[2]*y[1]^2 +
x[1]*x[3]*y[1]^2 - 4*x[1]^3*y[1]^3 - 3*x[1]^2*y[1]^4 -
x[1]*y[1]^5 - x[2]^2*y[2] + 2*x[1]*x[2]*y[1]*y[2] -
x[1]^2*y[1]^2*y[2] - x[2]*y[2]^2 + x[1]*y[1]*y[2]^2 -
x[3]*y[3] + x[1]^2*y[1]*y[3] + x[1]*y[1]^2*y[3] + y[6]

(6.13) The formulas (6.8) by computer.

X[1] = x[1]*y[1]
X[2] = x[2]*y[1]^2 + x[1]^2*y[2] + 2*x[2]*y[2]
X[3] = x[3]*y[1]^3 + x[1]^3*y[3] + 3*x[3]*y[3]
X[4] = x[4]*y[1]^4 - x[1]^2*x[2]*y[1]^2*y[2] -
2*x[2]^2*y[1]^2*y[2] - 2*x[1]^2*x[2]*y[2]^2 -
x[2]^2*y[2]^2 + 2*x[4]*y[2]^2 + x[1]^4*y[4] +
2*x[2]^2*y[4] + 4*x[4]*y[4]
X[5] = x[5]*y[1]^5 + x[1]^5*y[5] + 5*x[5]*y[5]
X[6] = x[6]*y[1]^6 - x[1]^2*x[2]^2*y[1]^4*y[2] -
2*x[2]^3*y[1]^4*y[2] - x[1]^4*x[2]*y[1]^2*y[2]^2 -
4*x[1]^2*x[2]^2*y[1]^2*y[2]^2 - 4*x[2]^3*y[1]^2*y[2]^2 -
2*x[1]^4*x[2]*y[2]^3 - 4*x[1]^2*x[2]^2*y[2]^3 -
2*x[2]^3*y[2]^3 + x[3]^2*y[2]^3 + 2*x[6]*y[2]^3 -
x[1]^3*x[3]*y[1]^3*y[3] - 3*x[3]^2*y[1]^3*y[3] +
x[2]^3*y[3]^2 - 3*x[1]^3*x[3]*y[3]^2 - 3*x[3]^2*y[3]^2 +
3*x[6]*y[3]^2 + x[1]^6*y[6] + 2*x[2]^3*y[6] + 3*x[3]^2*y[6] +
6*x[6]*y[6]

(6.14) The formulas (6.9) by computer.

X[1] = -x[1]
X[2] = -x[1]^2 - x[2]
X[3] = -x[3]
X[4] = -x[1]^4 - x[1]^2*x[2] - x[2]^2 - x[4]
X[5] = -x[5]
X[6] = x[1]^4*x[2] + x[1]^2*x[2]^2 - x[3]^2 - x[6]

7. Idempotents and products of rings.

My goal in the rest of these notes is to determine the structure of Λ(ℤ/pℤ), p a prime, and to relate this to the “Witt vectors” that first arise over p [1, page 175], [3]. This requires the study of idempotents in Λ(A).

I have assumed familiarity with products of rings. The relationship between product decompositions and idempotents may be less familiar so I will discuss that briefly now. Let R_{1} and R_{2} be rings (commutative with unit, as always). Then the product ring R = R_{1}ΠR_{2} is the set of ordered pairs (r_{1}, r_{2}), r_{1} ∈ R_{1}, r_{2} ∈ R_{2} with co-ordinatewise addition and multiplication. Let e_{1} = (1, 0) and e_{2} = (0, 1). It is easily checked that e_{1}^{2} = e_{1}, e_{2}^{2} = e_{2}, e_{1} + e_{2} = (1, 1) (which is the unit of R), and e_{1}e_{2} = 0. Furthermore the ideal Re_{1} of R is a ring which is isomorphic to R_{1} (also Re_{2} ≅ R_{2}). Similarly the product ring R_{1}ΠR_{2}Π … ΠR_{n} (also written Π_{i=1}^{n}R_{i}) is the set of n-tuples (r_{1}, …, r_{n}), r_{i} ∈ R_{i} with co-ordinatewise operations. More generally one can define the product Π_{i∈X}R_{i} over an arbitrary index set X. Previously we had the product Π_{i=1}^{∞}A over the infinite set ℕ.

Now let R be any commutative ring. An element e ∈ R such that e^{2} = e is called an idempotent. Every R contains the idempotents 0 and 1. Any other idempotent is referred to as non-trivial. Suppose that R contains a non-trivial idempotent e. Then f = 1 − e is also a non-trivial idempotent, and ef = 0. The ideals Re and Rf are rings (with unit), and the map r → (re, rf) induces an isomorphism of commutative rings R ≅ ReΠRf. More generally a finite set of idempotents {e_{1}, …, e_{n}} is called orthogonal if e_{i}e_{j} = 0 for i ≠ j and Σ_{i=1}^{n}e_{i} = 1. The homomorphism R → Π_{i=1}^{n}Re_{i} defined by r → (re_{1}, …, re_{n}) is an isomorphism.

Aside from finite products we will work also with the infinite product Π_{i=1}^{∞}A. Here we have idempotents δ_{i} (i ∈ ℕ), namely δ_{i} has i^{th} co-ordinate equal to 1 and all other co-ordinates equal to 0. We can add an infinite number of elements f_{j} ∈ Π_{i=1}^{∞}A as long as in each co-ordinate we add only a finite number of elements of A. For example we will write Σ_{i=1}^{∞}δ_{i} = 1, although we should remember that this is not the addition of the ring structure of Π_{i=1}^{∞}A. (Indeed when we write Π_{i=1}^{∞}A ≅ tA[[t]] we are implicitly permitting such infinite addition.) Similarly we can form infinite products as long as in each coordinate all but a finite number of the elements being multiplied are 1 (or one element is 0). The ring Π_{i=1}^{∞}A has lots of idempotents. Namely let S be any subset of ℕ. Let ϵ_{S} be the element of Π_{i=1}^{∞}A which has 1 in the i^{th} co-ordinate if i ∈ S, and 0 if i ∉ S. Clearly ϵ_{S} is an idempotent of Π_{i=1}^{∞}A. We have ϵ_{ℕ} = 1 and ϵ_{∅} = 0. For other subsets S ⊆ ℕ, ϵ_{S} is non-trivial. Any partition of ℕ as a finite disjoint union gives a finite set of orthogonal idempotents and hence a product decomposition of Π_{i=1}^{∞}A. For example ℕ = E ∐ O, where E is the set of even integers and O is the odd integers. This yields a product decomposition Π_{i=1}^{∞}A ≅ (Π_{ieven}A)Π(Π_{iodd}A).

If ℚ ⊆ A we have our isomorphism gh : Λ(A) → Π_{i=1}^{∞}A. If e is an idempotent of Π_{i=1}^{∞}A then gh^{−1}(e) is an idempotent of Λ(A). In particular Λ(ℚ) has lots of idempotents. But do any of these idempotents live in Λ(ℤ) (or in geometric language, is Spec(Λ(ℤ)) disconnected)? It follows from [1, Appendix to Lecture 26, C] that in fact Λ(ℤ) has no non-trivial idempotents, but I will not pursue this further. However we shall soon see that if appropriate integers are invertible in A, then Λ(A) does have non-trivial idempotents.

We have seen in (6.4) that gh(1−x_{n}t^{n}) = n(x_{n}t^{n}+x_{n}^{2}t^{2n}+x_{n}^{3}t^{3n}+…) and in particular that gh(1−t^{n}) = n(t^{n} +t^{2n} +t^{3n} +…) . If S is the subset of ℕ consisting of multiples of n, then we will write ϵ_{n} instead of ϵ_{S}. Thus we have gh(1−t^{n}) = n(t^{n} +t^{2n} +t^{3n} +…) = nϵ_{n}. Except for the n, we have that gh(1 − t^{n}) is an idempotent. If (1 − t^{n})^{1/n} can be defined in Λ(A) then we will have gh(1 − t^{n})^{1/n} = ϵ_{n}, at least if n is a non-zero-divisor in A, which leads us to expect that (1 − t^{n})^{1/n} will be an idempotent of Λ(A). If f ∈ Λ(A), by f^{1/n} (n^{th} root of f) we mean an element g ∈ Λ(A) such that g^{n} = f. Suppose that n is a unit in A (of course, by n ∈ A we mean the sum 1 + 1 + … + 1 of n 1’s). I claim then that f ∈ Λ(A) has a unique n^{th} root (equivalently n is a unit in Λ(A)). For suppose that f = 1+a_{1}t+a_{2}t^{2} +a_{3}t^{3} +…+a_{n}t^{n} +… and that g = 1+b_{1}t+b_{2}t^{2} +b_{3}t^{3} +…+b_{n}t^{n} +… is the proposed n^{th} root of f. Then by comparing coefficients of t^{i} in the equation f = g^{n} we obtain a system of equations of the form a_{1} = nb_{1}, …, a_{n} = nb_{n} + (lower b’s) which we can solve recursively (and uniquely) for the b_{i} in terms of the a_{i}. We can also find g by the binomial series. Write f = 1 + x, where x ∈ tA[[t]]. Then f^{1/n} = (1 + x)^{1/n} = 1 + Σ_{i=1}^{∞}\binom{1/n}{i}x^{i}. The binomial coefficient \binom{1/n}{i} is defined to be \frac{\frac{1}{n} ( \frac{1}{n} −1)...( \frac{1}{n} −i+1)}{i!} . This rational number can be written in the form \frac{a}{n^{d}} for a ∈ ℤ and d an integer ≥ 0, which then can be interpreted as an element of A by replacing 1/n by the (necessarily unique) inverse of n in A. That \binom{1/n}{i} has the indicated form is not obvious. You might try to prove this as an exercise. For example \binom{1/3}{4} = \frac{(1/3)(−2/3)(−5/3)(−8/3)}{1.2.3.4} = −10/3^{5}. The sum Σ_{i=1}^{∞}\binom{1/n}{i}x^{i} makes sense because x ∈ tA[[t]], so the coefficient of any power of t is a finite sum. We note that the integer “n” in Λ(A) is the series (1 − t)^{n} (1 − t being the unit of Λ(A)). The uniquely defined series (1 − t)^{1/n} will be denoted “1/n” and is the multiplicative inverse of “n” in Λ(A) (i.e. (1 − t)^{n} ∗ (1 − t)^{1/n} = 1 − t, which can be proved by the distributive law). Furthermore (1 − t)^{1/n} ∗ f = f^{1/n}. We now have (assuming n is a unit in A) that (1 − t^{n})^{1/n} does live in Λ(A) and that gh(1 − t^{n})^{1/n} = ϵ_{n}. From (6.10) we have (1 − t^{n}) ∗ (1 − t^{n}) = (1 − t^{n})^{n} from which it follows, using the distributive law, that (1 − t^{n})^{1/n} ∗ (1 − t^{n})^{1/n} = (1 − t^{n})^{1/n} so e_{n} := (1 − t^{n})^{1/n} is indeed idempotent in Λ(A). (I know that this is confusing. Try translating into the usual ring notation of + and multiplication.)

Finally we see that Λ(ℤ) is not an integral domain. For we have (1 − t^{n}) ∗ (1 − t^{n}) = (1 − t^{n})^{n} = (1 − t^{n}) ∗ (1 − t)^{n} (the first equality is (6.10), the second is multiplication by “n”). Equivalently we have (1 − t^{n}) ∗ [\frac{1−t^{n}}{(1−t)^{n}}] = 1 (1 being the “zero” element of Λ(A)). Clearly in Λ(ℤ) we have 1 − t^{n} ≠ (1 − t)^{n} if n > 1, so Λ(ℤ) is not an integral domain.

Now let S = {1, p, p^{2}, p^{3}, …}, p a prime number, and let T be the set of all primes but p. I will write ϵ_{<p>} for ϵ_{S} in this case. Then in Π_{n=1}^{∞}A, we have

(7.1)  ϵ_{<p>} = Π_{q∈T} (1 − ϵ_{q}) (= t + t^{p} + t^{p^{2}} + t^{p^{3}} + …)

Both sides of the equal sign are idempotent and have w_{i} = 0 if i is divisible by some prime in T, 1 if not. To give an explicit example take p = 3. Then in the first 11 co-ordinates we have

1 − ϵ_{2} = 1 0 1 0 1 0 1 0 1 0 1 …
1 − ϵ_{5} = 1 1 1 1 0 1 1 1 1 0 1 …

1 − ϵ_{7} = 1 1 1 1 1 1 0 1 1 1 1 …
1 − ϵ_{11} = 1 1 1 1 1 1 1 1 1 1 0 …
Π_{q∈T} (1 − ϵ_{q}) = 1 0 1 0 0 0 0 0 1 0 0 …

Subsequent q ∈ T will not change the first 11 co-ordinates of Π_{q∈T} (1 − ϵ_{q}). We see that Π_{q∈T} (1 − ϵ_{q}) has 1 in co-ordinates 1, 3, 3^{2}, … and 0 elsewhere.

Now assume that ℚ ⊆ A and continue to let S = {1, p, p^{2}, p^{3}, …}. Write ϵ_{<p>} = (w_{1}, w_{2}, …). Then gh^{−1}(ϵ_{<p>}) is an idempotent of Λ(A). Write gh^{−1}(ϵ_{<p>}) = 1 + a_{1}t + a_{2}t^{2} + … where a_{i} ∈ ℚ. I claim that a_{i} ∈ ℤ[T^{−1}] where by ℤ[T^{−1}] I mean that we invert the primes in T, or equivalently a_{i} = b_{i}/c_{i} where b_{i}, c_{i} ∈ ℤ, p ∤ c_{i}, i ≥ 1. (ℤ[T^{−1}] is the local ring ℤ_{(p)} of ℤ at the prime (p).) To see the claim note that a_{n} depends only on w_{i} (i ≤ n), hence a_{n} is the coefficient of t^{n} in gh^{−1}(Π_{1≤q≤n, q∈T} (1 − ϵ_{q})) = ∗_{q∈T,q≤n}[\frac{1−t}{(1−t^{q})^{1/q}}] and we have seen that the latter is of the form 1 + a′_{1}t + a′_{2}t^{2} + … where a′_{i} ∈ ℤ[T^{−1}] . We conclude that

(7.1.2)  E_{<p>} := gh^{−1}(ϵ_{<p>})

lives in Λ(A) whenever all primes but p are invertible in A and is the canonical image of an element of Λ(ℤ[T^{−1}]) which we might denote by ∗_{q∈T} [\frac{1−t}{(1−t^{q})^{1/q}}] . This infinite product in fact converges in an appropriate topology on Λ(A) or on Λ(ℤ[T^{−1}]). Furthermore gh(E_{<p>}) = ϵ_{<p>} whenever all primes in T are invertible in A.

Continue to assume that all primes in T are invertible in A and that S = {1, p, p^{2}, p^{3}, …}. Then E_{<p>} ∗ Λ(A) is a ring which is contained in Λ(A) (although with identity element E_{<p>} rather than the unit element 1 − t of Λ(A) ). I will write Λ_{<p>}(A) for Λ_{S}(A) if S = {1, p, p^{2}, p^{3}, …}. (This ring is usually called W (A) in the literature, for example (3, section 4), but I prefer a notation including the prime p.) I claim that under these hypotheses the map

(7.2)  E_{<p>} ∗ Λ(A) → Λ_{<p>}(A).

induced by the truncation homomorphism π_{<p>} : Λ(A) → Λ_{<p>}(A) is an isomorphism of rings.

Proof of (7.2): (1) If ℚ ⊆ A we have an isomorphism gh : Λ(A) → Π_{i=1}^{∞}A and gh(E_{<p>}) = ϵ_{<p>} so E_{<p>} ∗ Λ(A) → ϵ_{<p>}Π_{i=1}^{∞}A is an isomorphism of rings.

(2) Let ℘ : Π_{i=1}^{∞}A → ϵ_{<p>}Π_{i=1}^{∞}A be the canonical projection. For all A we have a commutative diagram of ring homomorphisms (7.3)

Λ(A) ⟶^{℘∘gh} ϵ_{<p>}Π_{i=1}^{∞}(A)
π_{<p>} ↓ ↗ τ
Λ_{<p>}(A)

where ℘∘gh factors through Λ_{<p>}(A) because the w_{i} (i ∈ S) depend only on the x_{i} (i ∈ S). Furthermore π_{<p>} is always onto.

(3) If all primes in T are invertible in A then (7.3) restricts to (7.4)

E_{<p>} ∘ Λ(A) ⟶^{℘∘gh} ϵ_{<p>}Π_{i=1}^{∞}(A)
π_{<p>} ↓  ↗ τ
Λ_{<p>}(A)

where π_{<p>}(E_{<p>}) = (1, 0, 0, 0, …) (the unit element of Λ_{<p>}(A)). The this follows from gh(E_{<p>}) = ϵ_{<p>} by solving the equations (6.6) over ℚ for x_{1}, x_{p}, x_{p^{2}}, …, obtaining x_{1} = 1, x_{p} = 0, x_{p^{2}} = 0, . . .. The latter hold in any A since E_{<p>} lives in Λ(ℤ[T^{−1}]). Thus π_{<p>} remains a surjective ring homomorphism.

(4) If in addition ℚ ⊆ A the top arrow of (7.4) becomes the isomorphism of part (1). By diagram chasing in (7.4) we then have that π_{<p>} is an isomorphism. The inverse η to π_{<p>} is defined by lifting f ∈ Λ_{<p>}(A) to Λ(A) in any way, and then multiplying by E_{<p>}. To define η in a definite way, lift (x_{1}, x_{p}, …) ∈ Λ_{<p>}(A) to (1 − x_{1}t)(1 − x_{p}t^{p}) … and then * with E_{<p>}. This calculation shows that η is defined by universal polynomials over ℤ[T^{−1}].

(5) Because η and E_{<p>}∗ are both defined by universal polynomials over ℤ[T^{−1}] the checking that η is a ring homomorphism and an inverse to π_{<p>} works assuming only that the primes of T are invertible in A. Thus (7.3) is an isomorphism whenever the primes of T are invertible in A, as we wished to show. (But note that ℘ ∘ gh may not be an isomorphism unless p is invertible in A).

8. Λ(A) in characteristic p.

Now we assume that A ≠ 0 is of characteristic p > 0. That is, p is a positive prime, and in A the sum 1+1+…+1 (p times) is zero. We will continue to denote the truncation set {1, p, p^{2}, ….} by S, and Λ_{S}(A) by Λ_{<p>}. In Λ(A) we have (1 − t)^{p} = 1 − t^{p} so the element “p” (sum of p 1’s in Λ(A) or Λ_{<p>}(A)) has x_{p} = 1 and x_{i} = 0 for i ≠ p. In other words

“p” = (0, 1, 0, 0, 0, …) ∈ Λ_{<p>}(A).

Similarly “p^{i}”, the sum of p^{i} 1’s in Λ(A) or Λ_{<p>}(A) (equal also to the i^{th} power of “p” in the ring structure of Λ(A) or Λ_{<p>}(A)) is equal to (0, 0, …, 1, 0, …) where the 1 is in the p^{i} position.

Now consider the formulae (6.10). Suppose i ≤ j. Then LCM(p^{i}, p^{j}) = p^{j} and GCD(p^{i}, p^{j}) = p^{i}. Thus (6.10) becomes

(8.1)  (1 − x_{p^{i}}t^{p^{i}}) ∗ (1 − x_{p^{j}}t^{p^{j}}) = (1 − x_{p^{i}}^{p^{j−i}}x_{p^{j}}t^{p^{j}})^{p^{i}} = (1 − x_{p^{i}}^{p^{j}}x_{p^{j}}^{p^{i}}t^{p^{i+j}})

In particular, taking i = 1, x_{p} = 1 we obtain (in Λ(A))

“p” ∗ (1 − xt^{p^{j}}) = (1 − t^{p}) ∗ (1 − xt^{p^{j}}) = (1 − x^{p}t^{p^{j+1}}). Thus in Λ_{<p>}(A) we have

(8.2)  “p” ∗ (x_{1}, x_{p}, x_{p^{2}} …) = (0, x_{1}^{p}, x_{p}^{p}, x_{p^{2}}^{p} …)

i.e. multiplication by “p” shifts right and raises to the p^{th} power. Now I claim

(8.3) Lemma. Suppose that A is an integral domain of characteristic p. Then Λ_{<p>}(A) is an integral domain of characteristic 0. Proof. Let f, g ∈ Λ_{<p>}(A) be non-zero. Write f = (0, 0, 0, …., x_{p^{i}}, ∗) and g = (0, 0, 0, …., y_{p^{j}}, ∗) where x_{p^{i}} and y_{p^{j}} are non-zero elements of A. Then by (8.1) we have that the product of f and g in Λ_{<p>}(A) is equal to (0, 0, …, 0, x_{p^{i}}^{p^{j}}y_{p^{j}}^{p^{i}}, ∗ …) where x_{p^{i}}^{p^{j}}y_{p^{j}}^{p^{i}} is in the p^{i+j} co-ordinate. Since A is an integral domain x_{p^{i}}^{p^{j}}y_{p^{j}}^{p^{i}} ≠ 0, and we conclude that Λ_{<p>}(A) is an integral domain. We already know that “n” is invertible in Λ_{<p>}(A) if GCD(n, p) = 1. Thus Λ_{<p>}(A) must either be of characteristic p or characteristic 0. By (8.2) we cannot have characteristic p, so Λ_{<p>}(A) is of characteristic 0.

(8.4) Lemma. Let A be a field of characteristic p. For f ∈ Λ_{<p>}(A), f ≠ 0, define v(f) := i where i is the smallest natural number such that the p^{i}-th co-ordinate of f is non-zero. Then we have (writing the ring operations of Λ_{<p>}(A) in the usual way)

(8.4.1)  v(fg) = v(f) + v(g)
(8.4.2)  v(f + g) ≥ min(v(f), v(g))

Proof. Property (8.4.1) follows from the proof of (8.3). Property (8.4.2) follows from the universal polynomials (6.7). The p^{i} co-ordinate of f + g depends only on the co-ordinates p^{j}(j ≤ i) of f and g, and if all the latter co-ordinates are 0, so is the p^{i} co-ordinate of f + g, from which (8.4.2) follows.

(8.5) Lemma. Let A be any field of characteristic p. Then Λ_{<p>}(A) is a local ring with maximal ideal M = {(0, ∗, ∗ …)} = {f ∈ Λ_{<p>}(A) : v(f) > 0} (all elements of Λ_{<p>}(A) with first co-ordinate 0). Proof. The surjection Λ_{<p>}(A) → A given by projection (x_{1}, x_{p}, …) → x_{1} is a ring homomorphism because the first universal polynomial in (6.7) and in (6.8) are z_{1} = x_{1} +y_{1} and z′_{1} = x_{1}y_{1} respectively. This shows that M is a maximal ideal. We will complete the proof of (8.5) by showing that if g ∈ Λ_{<p>}(A) and v(g) = 0, then g is a unit in Λ_{<p>}(A). Let g = (x_{1}, x_{p}, …) with x_{1} ≠ 0. In the universal polynomials (6.8) I claim that z′_{p^{n}} = x_{1}^{p^{n}}y_{p^{n}} + terms in y_{p^{i}} with i < n. Given this claim, we can recursively solve for the co-ordinates y_{p^{i}} in the equation g ∗ (y_{1}, y_{p}, …) = (1, 0, 0, 0, …) from which we conclude that g is a unit in Λ_{<p>}(A) and (8.5) follows. To see the claim, lift to Λ(A): we have

[(1 − x_{1}t)(1 − x_{p}t^{p})(1 − x_{p^{2}}t^{p^{2}}) …] ∗ [(1 − y_{1}t)(1 − y_{p}t^{p})(1 − y_{p^{2}}t^{p^{2}}) …]

Multiply this out using (8.1) and the distributive law of Λ(A). The lowest power of t that contains y_{p^{n}} in its coefficient is t^{p^{n}}, from (1 − x_{1}t) ∗ (1 − y_{p^{n}}t^{p^{n}}) = 1 − x_{1}^{p^{n}}y_{p^{n}}t^{p^{n}}. Other contributions to the coefficient of t^{p^{n}} involve only y_{p^{i}} for i < n, so z′_{p^{n}} is as claimed, completing the proof.

(8.6) Lemma. Let A be a perfect field of characteristic p. Then the only non-zero ideals of Λ_{<p>}(A) are the powers M^{i} = {(0, 0, …, 0, ∗ ….)} (with non-zero elements only in co-ordinates p^{i} or greater), and M^{i} is generated by “p^{i}”.

Proof. Let I be a non-zero ideal of Λ_{<p>}(A). Let i = min(v(f) | f ∈ I, f ≠ 0}. Let h ∈ I with v(h) = i. By (8.2), (using the assumption that A is perfect which implies that p^{i}-th roots exist in A for all i ≥ 0) we have that h=“p^{i}”∗g where v(g) = 0. By (8.5) g is a unit so “p^{i}” ∈ I. From this it follows that “p^{i}” generates I, that I = M^{i}, and M^{i} is as described in the statement of the Lemma.

(8.7) Theorem. If A is a perfect field of characteristic p, and S = {p^{i}}_{i≥0} then Λ_{<p>}(A) := Λ_{S}(A) is a complete discrete valuation ring of characteristic 0, with maximal ideal M generated by “p”. Proof. In view of (8.3), (8.5), (8.6) and a suitable characterization of discrete valuation ring (e.g. Proposition 9.2 of Atiyah-Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969) we have that Λ_{<p>}(A) is a discrete valuation ring of characteristic 0 with maximal ideal M generated by “p”, and associated valuation v as in (8.4). We need only show that Λ_{<p>}(A) is complete in the valuation v. Let f = (a_{1}, a_{p}, a_{p^{2}}, …), g = (b_{1}, b_{p}, b_{p^{2}}, . . . ..). I claim that v(f − g) = i where the first co-ordinate in which f and g differ is the p^{i}-th. To see this lift to Λ(A). The difference f − g in Λ_{<p>}(A) lifts to F = [(1 − a_{1}t)(1 − a_{p}t^{p})(1 − a_{p^{2}}t^{p^{2}}) …]/[(1 − b_{1}t)(1 − b_{p}t^{p})(1 − b_{p^{2}}t^{p^{2}}) …]. If f and g differ first in co-ordinate p^{i}, terms in t^{p^{j}} cancel for j < i, and F = (1 − (a^{p^{i}} − b^{p^{i}})t^{p^{i}}) (terms involving only higher powers of t). This proves the claim. From this we see that a sequence \{\underline{x}_{1}, \underline{x}_{2}, … \underline{x}_{n} …\} in Λ_{<p>}(A) where \underline{x}_{i} = (x_{i1}, x_{ip}, …) i ≥ 1) is a Cauchy sequence if and only if each sequence {x_{ip^{j}}} (j fixed, i varying) stabilizes at some value y_{p^{j}} as i → ∞. Clearly \{\underline{x}_{1}, \underline{x}_{2}, … \underline{x}_{n} …\} converges to the element (y_{1}, y_{p}, y_{p^{2}} …) ∈ Λ_{<p>}(A) completing the proof of (8.7).

We remark that if A is a field of characteristic p, but not perfect, then M^{i} is strictly smaller than {f ∈ Λ_{<p>}(A) : v(f) ≥ i} and Λ_{<p>}(A) is not a valuation ring. The first observation follows from (8.1), which implies, for example, that every element of M^{2} is of the form (0, 0, a, …) with a ∈ A^{p}. If x is an element of the quotient field of Λ_{<p>}(A), then either x or 1/x (the multiplicative inverse of x under ∗) is in Λ_{<p>}(A). Consider x = c/“p”, where c is of the form (0, 0, a, …) with a ∉ A^{p}. Then v(x) = 1 so 1/x cannot be in Λ_{<p>}(A) since all elements y of the latter have v(y) ≥ 0. But also x ∉ Λ_{<p>}(A), by the above discussion.

For convenience of notation, in the rest of this section I will denote an arbitrary element of Λ_{<p>}(A) by (x_{0}, x_{1}, …,) so that what we formerly called x_{p^{i}} will now be denoted by x_{i}, i ≥ 1. The map f : A → Λ_{<p>}(A) defined by f(a) = (a, 0, 0, …) preserves multiplication but not addition (6.7), (6.8). If we write the operations of Λ_{<p>}(A) as “+” and juxtaposition instead of respectively juxtaposition and * we can (by the discussion after (6.9) ) write x = (x_{0}, 0, 0, 0, . . . ..) + (0, x_{1}, 0, 0, . . . ..) + (0, 0, x_{2}, . . . ..) + . . .. Applying (8.2) to the latter we get x = f(x_{0}) + f(x_{1})^{p^{−1}}“p”+f(x_{2})^{p^{−2}}“p^{2}” + …. Now let A = ℤ/pℤ and let ℤ̂_{p} denote the p-adic integers. The expression for x simplifies to x = f(x_{0}) + f(x_{1})“p”+f(x_{2})“p^{2}” + …. Every element a of ℤ̂_{p} has a unique expansion in the form a = a_{0} + a_{1}p + a_{2}p^{2} + … where each a_{i} is either 0 or a primitive (p − 1)^{st} root of unity in ℤ̂_{p}. The p possible values for the coefficients a_{i} map under reduction mod p bijectively onto the elements of ℤ/pℤ, say a_{i} → ā_{i}. The unique ring homomorphism ℤ → Λ(ℤ/pℤ) is an inclusion by (8.3). This extends to a unique homomorphism ℤ_{(p)} → Λ(ℤ/pℤ) since all primes but p are invertible in Λ(ℤ/pℤ), and then by continuity to an inclusion ρ : ℤ̂_{p} → Λ(ℤ/pℤ) . Since a_{i} is a (p − 1)^{st} root of unity so is ρ(a_{i}) and since ρ commutes with the “mod p” maps to ℤ/pℤ we have that ρ(a_{i}) = f(ā_{i}). Thus we have ρ(a) = ρ(a_{0} + a_{1}p + a_{2}p^{2} + …) = ρ(a_{0}) + ρ(a_{1})“p” +ρ(a_{2})“p^{2}” + … = f(ā_{0}) + f(ā_{1})“p” +f(ā_{2})“p^{2}” + . . .. Since the a_{i} are arbitrary (p − 1)^{st} roots of unity, ρ is onto, hence an isomorphism.

More generally if A is any perfect field of characteristic p then Λ(A) is the unique (up to canonical isomorphism) discrete valuation ring of characteristic 0 such that v(p) = 1 and Λ(A)/pΛ(A) ≅ A (the proof of uniqueness can be found in [6, II §5]). If R is such a discrete valuation ring then there is a map f : R/pR → R that preserves multiplication (so that f(0) = 0 and if a ≠ 0 f(a) has p^{i}-th roots for all i). Every element a ∈ R has a unique expansion a = a_{0} + a_{1}^{p^{−1}}p + a_{2}^{p^{−2}}p^{2} + … where a_{i} = f(ā_{i}). The map ρ : R → Λ_{<p>}(A) defined by ρ(a) = f(ā_{0}) + f(ā_{1})^{p^{−1}} “p”+f(ā_{2})^{p^{−2}}“p^{2}”+ …, (which corresponds to (ā_{0}, ā_{1}, ā_{2}, …) in the vector notation used above for Λ_{<p>}(A)) is an isomorphism ( page 49 of [6] and page 175 of [1]).

9. The Verschiebung and the Frobenius.

In this final section I will describe the complete structure of Λ(ℤ/pℤ). In order to do this it is helpful to introduce two homomorphisms V_{n}, F_{n} : Λ(A) → Λ(A) (A any commutative ring, n ∈ ℕ).

Define V_{n} : Λ(A) → Λ(A) by V_{n}(f(t)) = f(t^{n}). This is a homomorphism of abelian groups, but not of rings if n > 1. If f = (1 − x_{1}t)(1 − x_{2}t^{2})(1 − x_{3}t^{3}) …, then V_{n}(f) = (1 − x_{1}t^{n})(1 − x_{2}t^{2n})(1 − x_{3}t^{3n}) … so in Λ_{ℕ}(A) (≅ Λ(A)) V_{n} acts by V_{n}(x_{1}, x_{2}, x_{3}, …) = (x′_{1}, x′_{2}, x′_{3}, …) where

(9.1) x′_{m} = x_{m/n} if n | m and 0 otherwise.

For example V_{2}(x_{1}, x_{2}, x_{3}, …) = (0, x_{1}, 0, x_{2}, 0, x_{3}, …).

Suppose that gh(f(t)) = w_{1}t + w_{2}t^{2} + …. Then gh(f(t^{n})) = −nt^{n}f′(t^{n})/f(t^{n}) = n(w_{1}t^{n} + w_{2}t^{2n} + …), so if gh(V_{n}(f)) = w′_{1}t + w′_{2}t^{2} + … then

(9.2) w′_{m} = nw_{m/n} if n | m and 0 otherwise.

Note that ℕ−nℕ (all natural numbers not divisible by n) is a truncation set. (Clearly if n does not divide a, then n does not divide any divisor of a.) The homomorphism V_{n} is injective, and the image of V_{n} is the kernel of the truncation map Λ_{ℕ}(A) → Λ_{ℕ−nℕ}(A). Obviously

(9.3)  V_{m} ∘ V_{n} = V_{mn}.

The map F_{n} is somewhat more subtle. In the w co-ordinates F_{n} turns out to be fairly simple, but since gh is not an isomorphism one has to be careful. In order to define F_{n} I will initially assume that the complex numbers ℂ ⊆ A. Define F_{n} : Λ(A) → Λ(A) by

(9.4)  F_{n}(f) = Π_{ζ}(f(ζt^{1/n}))

where ζ ranges over the n^{th} roots in ℂ of 1 (so that t^{n} − 1 factors as Π(t − ζ)). Replacing t by t^{−1/n} and multiplying by t we have Π_{ζ}(1 − ζt^{1/n}) = 1 − t. To see that the right hand side of (9.4) lies in Λ(A) it suffices to check that F_{n}(1−x_{m}t^{m}) lies in Λ(A). As in section 6 we will write M =GCD(m, n) and N =LCM(m, n). We have that ζ_{1}^{m}, …, ζ_{n}^{m} are the roots (repeated M times) of x^{n/M} − 1. Let η range over the n/M^{th} roots of 1. It now follows that F_{n}(1 − x_{m}t^{m}) = Π_{ζ}(1 − x_{m}ζ^{m}t^{m/n}) = (Π_{η}(1 − x_{m}ηt^{m/n}))^{M} = (1 − x_{m}^{n/M}t^{m/M})^{M}. From mn = MN we have

(9.5)  F_{n}(1 − x_{m}t^{m}) = (1 − x_{m}^{N/m}t^{N/n})^{M} = (1 − x_{m}^{n/M}t^{m/M})^{M}

Applying (9.5) to f = 1 + a_{1}t + a_{2}t^{2} + … = (1 − x_{1}t)(1 − x_{2}t^{2})(1 − x_{3}t^{3}) … we see that F_{n}(f) ∈ Λ(A) indeed F_{n}(f) = 1 + Σ_{i≥1}p_{i}(x_{1}, x_{2}, …)t^{i}, where the p_{i} are polynomials with integer coefficients, independent of A. Furthermore we have (9.6) ∑_{i=1}^{n} ζ_{i}^{a} = n if a is a multiple of n and 0 otherwise.

We apply (9.6) to determine how F_{n} behaves on the w co-ordinates, where gh(f(t)) = w_{1}t + w_{2}t^{2} + …. For any λ we have gh(f(λt^{1/n})) = (1/n)(λt^{1/n}w_{1} + λ^{2}t^{2/n}w_{2} + λ^{3}t^{3/n}w_{3} + …). Letting λ range over the ζ_{i}, summing, and using (9.6) we obtain gh(F_{n}(f(t)) = w_{n}t + w_{2n}t^{2n} + w_{3n}t^{3n} + …. that is, if

gh(F_{n}(f(t))) = w′_{1}t + w′_{2}t^{2} + … then
(9.7)  w′_{m} = w_{mn}

The conditions (9.7) define a ring homomorphism Π_{i=1}^{∞}A → Π_{i=1}^{∞}A. Therefore F_{n} is a ring homomorphism.

Because the p_{i} mentioned after (9.5) have coefficients in ℤ we can use them to define F_{n} for arbitrary A. By the usual universal polynomial argument using the p_{i} we obtain that F_{n} is a ring homomorphism for all A.

We now examine three extreme cases of (9.5). First of all, assume that m and n are relatively prime. Then M = GCD(m, n) = 1, and (9.5) becomes

(9.8)  F_{n}(1 − x_{m}t^{m}) = (1 − x_{m}^{n}t^{m}), GCD(m, n) = 1

If n | m then N = m and M = n and we get

(9.9)  F_{n}(1 − x_{m}t^{m}) = (1 − x_{m}t^{m/n})^{n}, n | m

If m | n then N = n and M = m and we get

(9.10)  F_{n}(1 − x_{m}t^{m}) = (1 − x_{m}^{n/m}t)^{m}, m | n

As one example we have F_{2}(1−x_{m}t^{m}) = (1−x_{m}(t^{1/2})^{m})(1−x_{m}((−1)t^{1/2})^{m}). If m is even this equals (1 − x_{m}t^{m/2})^{2} as in (9.9). If m is odd we get (1 − x_{m}(t^{1/2})^{m})(1 + x_{m}(t^{1/2})^{m}) = 1 − x_{m}^{2}t^{m} as in (9.8).

We now have

(9.11)  F_{n} ∘ V_{n} = n

This can be checked using (9.9) or by observing that on the w co-ordinates F_{n} ∘ V_{n} = n (check first assuming that A contains ℚ and then use the universal argument). Note that (9.11) is written using additive notation, so that for f ∈ Λ(A), (9.11) means that F_{n}(V_{n}(f)) = f^{n}. If n is a unit in A (hence also in Λ(A), as was noted in Section 7) we have from (9.11) that F_{n} ∘ ((1/n)V_{n}) = 1 (the identity endomorphism) from which it follows that ((1/n)V_{n}) ∘ F_{n} is an idempotent endomorphism of Λ(A). In fact, we have ((1/n)V_{n}) ∘ F_{n}(1 − t) = ((1/n)V_{n})(1 − t) = (1 − t^{n})^{1/n} which was shown in Section 7 to be an idempotent of Λ(A). Furthermore (1/n)V_{n} is a homomorphism of rings without unit, as can be checked in the w co-ordinates if ℚ ⊆ A, hence holds whenever n is a unit in A.

As in Section 7 let S = {1, p, p^{2}, …} (p a prime). Suppose that GCD(q, p) = 1, q ∈ ℕ. Let qS = {q, qp, qp^{2}, …}. The idempotent giving projection of Π_{i=1}^{∞}A onto the co-ordinates qS is ϵ_{qS} = t^{q} + t^{qp} + t^{qp^{2}} + …. Clearly in Π_{i=1}^{∞}A, ϵ_{qS} = Π_{α}(ϵ_{q} − ϵ_{qα}) as α ranges over all primes except p. (As in Section 7, for n ∈ ℕ, ϵ_{n} = t^{n} + t^{2n} + t^{3n} + …, the idempotent of Π_{i=1}^{∞}A giving projection onto the co-ordinates that are multiples of n.) If all primes but p are invertible in A, we can define

(9.12) E_{<q,p>} := ∗_{α} \frac{(1−t^{q})^{1/q}}{(1−t^{qα})^{1/qα}} (α ranging over all primes except p). Then E_{<q,p>} is an idempotent of Λ(A) and gh(E_{<q,p>}) = ϵ_{qS}. We now have

(9.13) Lemma. If all primes but p are invertible in A, then Π_{q}E_{<q,p>} = 1−t (q ranging over all elements of ℕ that are prime to p). Furthermore if q_{1} ≠ q_{2} then E_{<q_{1},p>}∗E_{<q_{2},p>} = 1. (Here Π means multiplication of power series, i.e. the addition of Λ(A), 1 − t is the unit element, and 1 is the 0 element of Λ(A)), so we obtain an infinite product decomposition of Λ(A)). Proof. If ℚ ⊆ A we can apply the isomorphism gh. The assertion (9.13) becomes Σ_{q}ϵ_{qS} = 1 and ϵ_{q_{1}S}ϵ_{q_{2}S} = 0, both of which are clearly true, so (9.13) holds if ℚ ⊆ A. But the coefficients of the E_{<q,p>} live in ℤ_{(p)} (in the notation of Section 7, ℤ_{(p)} = ℤ[T^{−1}] .) Hence (9.13) holds for all A in which all primes but p are invertible.

Clearly we have ((1/q)V_{q})(E_{<p>}) = E_{<q,p>} (E_{<p>} is as in Section 7, and is equal to E_{<1,p>} in the notation of (9.12) ). Also we have

(9.14) Theorem. If all primes but p are invertible in A, then

((1/q)V_{q}) : E_{<p>} ∗ Λ(A) → E_{<q,p>} ∗ Λ(A)

is an isomorphism of rings (with inverse F_{q}) for all q prime to p. Proof. If ℚ ⊆ A we can apply the isomorphism gh and check in the w co-ordinates where the assertion is clear. The result then follows for all A in which all primes but p are invertible, since E_{<p>} and E_{<q,p>} both have coefficients in ℤ_{(p)}, (1/q)V_{q} is defined over ℤ[q^{−1}] ⊆ ℤ_{(p)} and F_{q} is defined over ℤ.

Putting (9.13) and (9.14) together we obtain our main result:

(9.15) Theorem. If all primes but p are invertible in A, then Λ(A) ≅ ΠΛ_{<p>}(A) (one copy for every q ∈ ℕ, GCD(q, p) = 1) , where S = {1, p, p^{2}, …} and Λ_{<p>}(A) is the truncation onto S as defined in (6.11) and Section 7. The product is taken over all natural numbers prime to p. In particular Λ(ℤ/pℤ) ≅ Πℤ̂_{p}.

I would like to make one final remark: (in the renumbering introduced in the second last paragraph of section 8) V_{p} acts on Λ_{<p>}(A) by V_{p}(x_{1}, x_{2}, …) = (0, x_{1}, x_{2}, …), so the name Verschiebung (which in german means shift) seems appropriate. Furthermore by (9.10) we have that F_{p}(1 − x_{i}t^{p^{i}}) = (1 − x_{i}^{p}t^{p^{i}}), i.e. F_{p} acts by raising the co-ordinates to the p^{th} power. This is consistent with the familiar meaning of Frobenius in algebraic geometry in characteristic p. Also (F_{p} ∘V_{p})(x_{1}, x_{2}, …) = (0, x_{1}^{p}, x_{2}^{p}, …) = “p”∗(x_{1}, x_{2} …), so F_{p} ∘ V_{p} = p agreeing with (9.11). On Λ_{<p>}(A) we also have V_{p} ∘ F_{p} = p.

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