Remark 1

We will see below (X, p. 107, Prop. 6) some finiteness properties of the modules $\operatorname{Ext}^{n}_{A}(M,N)$. For example, if $A$ is commutative and noetherian, and if $M$ and $N$ are $A$-modules of finite type, each $A$-module $\operatorname{Ext}^{n}_{A}(M,N)$ is of finite type.

Let $f : M' \to M$ and $g : N \to N’$ be homomorphisms of $A$-modules. Set

this is a homomorphism of degree 0 of graded $k$-modules

whose homogeneous components are denoted

According to Prop. 1 of [X], p. 82, the canonical homomorphism

is bijective; using the isomorphisms $M\to H_{0}(L(M))$ and $H^{0}(I(N))\to N$, one deduces a canonical isomorphism

(7)

We will always identify $\operatorname{Ext}^{0}_{A}(M,N)$ with $\operatorname{Hom}_{A}(M,N)$ via this isomorphism. Then the $k$-linear map $\operatorname{Ext}_{A}(f,g)$ can be identified with $\operatorname{Hom}_{A}(f,g)$.