Mathematics 894, Problem Set 2

Due January 25, 2008. Read Section 9.6 and Appendix I, section 2. Throughout, F is a field. You may use results from Problem Set 1. The hints are only a suggestion. You may be able to find some other approach.

(1) Let R = F [t, x, y] and I =< t^{2} −xy, t^{3} −y >. Use lexicographic order with t > x > y.

(a) Find (by hand) a Groebner basis for I. Use Buchberger’s criterion to check that you have a Groebner basis.

(b) Give a set of generators for LT (I).

(2) Let R = F [X_{1} … X_{n}]. Order the monomials in R by the coordinatewise partial order on ℕ^{n}, i.e. (a_{1}, a_{2}, …, a_{n}) ≤_{c} (b_{1}, b_{2}, …, b_{n}) if and only if a_{i} ≤ b_{i}, 1 ≤ i ≤ n. We will write <_{c} to mean ≤_{c} but not equal.

(a) Explain why ≤_{c} is not a total order if n ≥ 2.

(b) Let ℳ be any set of monomials in R (finite or infinite). Show that ℳ has a finite number of minimal elements under ≤_{c}. (A minimal element of ℳ is an element m ∈ ℳ such that there is no element n of ℳ with n <_{c} m.)

(c) Interpret (b) in terms of generating sets for monomial ideals in R.

Hint: define a suitable ideal from ℳ and use ring theoretic results.

(3) Let R = F [X, Y] and I =< X, Y >. Describe I/I^{2} both as an F -vector space and as an R-module.

(4) Let R = F [X, Y] and I =< X, Y >. Prove that X ⊗ Y − Y ⊗ X = 0 in R ⊗_{R} R ≅ R but that X ⊗ Y − Y ⊗ X ≠ 0 in I ⊗_{R} I. Hint: Define an R-bilinear map from I × I to somewhere (perhaps considering the previous question) under which (X, Y) and (Y, X) have different images. From this conclude that X ⊗ Y − Y ⊗ X ≠ 0 in I ⊗_{R} I.

Notes: We did not prove several Theorems in the text, in particular, Proposition 24, Lemma 25, Proposition 26 of section 9.6. You might read (or try to prove yourself) the proof of Proposition 24. For the other two results, in this course it suffices for you to understand the statement of Proposition 26.

You might start studying section 10.4, which does tensor product. In the lectures I will confine detailed proofs to the case where R is commutative. I am using the universal mapping property of Corollary 12, page 368, as the definition of M ⊗_{R} N and things like pages 360-361 as a construction of M ⊗_{R} N.