Exercise 8, § 8

Let $A$ be a noetherian local ring.

a) Let $M$ and $N$ be $A$-modules such that $\mathrm{dpl}_A(M) < +\infty$ and $\mathrm{Tor}_i^A(M, N) = 0$ for $i > 0$. Define for every integer $n$ a canonical isomorphism

and prove that one has $\mathrm{di}_A(M \otimes_A N) \leqslant \mathrm{di}_A(N)$ (adapt the proof of Exercise 6 of § 3 with the aid of Exercise 7).

b) Let $N$, $P$ be $A$-modules such that $\mathrm{di}_A(P) < +\infty$ and $\mathrm{Ext}_A^i(N, P) = 0$ for $i > 0$. Define for every integer $n$ a canonical isomorphism

(apply Matlis duality).

c) Prove that if there exists an $A$-module $M$ whose injective dimension and flat dimension are finite, $A$ is a Gorenstein ring (take $N = A$ in a)).

d) Conversely, if $A$ is a Gorenstein ring, prove that there is an identity between modules of finite injective dimension and modules of finite flat dimension (prove with the aid of $a$) that a module of finite flat dimension has finite injective dimension, and obtain the converse by Matlis duality).

e) Deduce from $d$ that the following conditions are equivalent: (i) $A$ is a Gorenstein ring ; (ii) the $A$-modules of finite type and of finite injective dimension coincide with the $A$-modules of finite type and of finite projective dimension ; (iii) there exists an $A$-module of finite type whose injective dimension and projective dimension are finite.