Remark 1

We will see below (X, p. 107, Proposition 6) finiteness properties of the modules $\operatorname{Tor}^A(M, N)$. For example, if $A$ is commutative noetherian and if $M$ and $N$ are $A$-modules of finite type, every $A$-module $\operatorname{Tor}_n^A(M, N)$ is of finite type.

Let $f : M \to M'$ be a homomorphism of right $A$-modules and $g : N \to N'$ a homomorphism of left $A$-modules. We define $\operatorname{Tor}^A(f, g) = H(L(f) \otimes_A L(g))$; this is a homomorphism of graded $k$-modules

whose homogeneous components are denoted

According to Proposition 1 of X, p. 62, the canonical homomorphism

is bijective; using the isomorphisms $M \to H_0(L(M))$ and $N \to H_0(L(N))$, we deduce from it an isomorphism, called canonical

(7)

We will always identify $\operatorname{Tor}_0^A(M, N)$ with $M \otimes_A N$ through this isomorphism. Hence the $k$-linear map $\operatorname{Tor}_0^A(f, g)$ is identified with $f \otimes g$.