Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

We recommend the book of Lang [1] as a good reference. The treatment here follows the book for the most part.

THEOREM 07.1   Let $\mathcal{C}_1$ be an abelian category with enough injectives, and let $F:\mathcal{C}_1\to \mathcal{C}_2$ be a covariant additive left functor to another abelian category $\mathcal{C}_2$ . Then:

1. $F\cong R^0 F$ .
2. For each short exact sequence
   $$0\to M'\to M \to M''\to 0$$
   and for each $n\geq 0$ there is a natural homomorphism
   $$\delta^n: R^n F(M'')\to R^{n+1} F(M)$$
   such that we obtain a long exact sequence
   $$\dots\to R^n F(M')\to R^n F(M) \to R^n F(M'')\overset{\delta^n}{\to} R^{n+1} F(M')\to \dots.$$

3. $\delta$ is natural. That means, for a morphism of short exact sequences
   $$\begin{CD} 0 @»> M'@»> M@»> M''@»> 0\\ @. @VVV @VVV @VVV \\ 0@»> N'@»> N@»> N''@»> 0 \end{CD}$$
   the $\delta$ 's give a commutative diagram:
   $$\begin{CD} R^n F(M'') @> \delta^n » R^{n+1}F(M') \\ @VVV @VVV \\ R^nF(N'') @> \delta^n » R^{n+1} F(N') \end{CD}$$

4. For each injective objective object $I$ of $A$ and for each $n>0$ we have $R^n F(I)$ .

LEMMA 07.2   For any exact sequence $0 \to M'\to M\to M''\to 0$ of objects in $\mathcal{C}_1$ , There exists injective resolutions $I_{M'},I_{M},I_{M''}$ of $M',M,M''$ respectively and a commutative diagram

$$\begin{CD} @. 0 @. 0 @. 0@.\\ @. @VVV @VVV @VVV \\ 0 @»> M'@»... ...@»> 0\\ @. @VVV @VVV @VVV \\ 0@»> I_{M'}@»> I_M@»> I_{M''}@»> 0 \end{CD}$$

such that the diagram of resolutions is exact.

DEFINITION 07.3   Let $F$ be a left exact additive functor. An object $X$ is called $F$ -acyclic if $R^n F(X)=0$ for all $n>0$ .

THEOREM 07.4   Let

$$0\to M \to X^0 \to X^1 \to X^2 \to \dots$$

be a resolution of $M$ by $F$ -acyclics. Let

$$0\to M \to I^0 \to I^1 \to I^2 \to \dots$$

be an injective resolution. Then there exists a morphism of complexes $X\to I$ extending the identity on $M$ , and this morphism induces an isomorphism

$$H^nF(X)\cong H^n (F(I))=R^n F(M)$$

for all $n\geq 0$ .

Note: Our notation of denoting complexes such as $I_M$ differs from that in [1].

The book of Grivel [2] is also a good reference for our future arguments.