Exercise 6, § 3

Commutative Algebra, X

Bourbaki marks this one as harder than the rest.

Let $A$ be a noetherian local ring, $M$ a finitely generated $A$ -module and of finite projective dimension, $N$ an $A$ -module such that Tor_i^A(M, N) = 0 for i > 0. (a) Define for every integer n a canonical isomorphism

\operatorname{Ext}_A^n(\kappa_A, M \otimes_A N) \longrightarrow \bigoplus_{p \geq n} \operatorname{Tor}_{p-n}^A(\kappa_A, M) \otimes_k \operatorname{Ext}_A^p(\kappa_A, N),

and prove that one has di_A(M \otimes_A N) \leq \operatorname{di}_A(N). (Let (P, p) be a finite free resolution of M, (I, e) an injective resolution of N, and C the complex P \otimes_A I. Prove that the complex

0 \to C^0/B^0(C) \to C^1 \to C^2 \to \ldots

defines an injective resolution of M \otimes_A N.) (b) Prove the inequality prof_A(N) = dp_A(M) + prof(M \otimes_A N). Deduce another proof of the theorem of Auslander-Buchsbaum. (c) Suppose in addition that N is of finite projective dimension. Prove the equality dp_A(M \otimes_A N) = dp_A(M) + dp_A(N). (d) Suppose that A is a Gorenstein ring, and denote its dimension by d. Let n be an integer; show that the $\kappa_A$ -vector spaces $\operatorname{Ext}_A^n(\kappa_A, M)$ , $\operatorname{Tor}_{d-n}^A(\kappa_A, M)$ and $\operatorname{Ext}_A^{d-n}(M, \kappa_A)$ have the same dimension.

Solution

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