Proposition 3

a) If $\mathrm{dp}_A(M) \leqslant n$, the canonical homomorphism (3) is injective. b) If $\mathrm{dp}_A(M) \leqslant n$, if $A$ is left noetherian and if $M$ is of finite type, the canonical homomorphism (3) is bijective.

We argue by induction on $n$. If $n = 0$, $M$ is projective, the homomorphism (3) reduces to the canonical homomorphism $P \otimes_A M \to \operatorname{Hom}_A(M^*, P)$ of II, p. 77, and the proposition follows from loc. cit., corollary. If $n > 0$, let

(4)

be an exact sequence of $A$-modules where $L$ is free (and of finite type in case b); then $\mathrm{dp}_A(N) \leqslant n - 1$ (X, p. 135, cor. 2, c)) and $N$ is of finite type in case b.

Let $\theta \in \operatorname{Ext}_A^1(M, N)$ be the class associated with the exact sequence (4) (X, p. 117, déf. 1). Denote

the mappings defined by $u_n(\alpha) = \alpha \circ \theta$ and $v_n(\beta) = \theta \circ \beta$. We have

for all $\alpha\in\operatorname{Ext}_A^{n-1}(N,A)$, $\beta\in\operatorname{Tor}_n^A(P,M)$ (X, p. 129, prop. 6), so that the diagram

where the horizontal arrows are the canonical homomorphisms, is commutative.

If $n=1$, we thus have a commutative diagram:

where the columns are exact; this yields the result in this case. If $n\geq 2$, the mappings $u_n$ and $v_n$ are bijective (X, p. 128, cor. 4 and p. 131, cor. 3). By the induction hypothesis, the canonical homomorphism

is injective (resp. bijective); the diagram (5) shows that the same holds for the canonical homomorphism $\operatorname{Tor}_n^A(P,M)\longrightarrow\operatorname{Hom}_A(\operatorname{Ext}_A^n(M,A),P)$, which completes the proof.