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 $R^n F(I)$ .

The collection $\{R^jF\}$ of functors $R^jF$ is a ``universal delta functor''. See [1].

LEMMA 07.2   Under the assumption of the previous theorem, 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. Thus we obtain a diagram

$$\begin{CD} @. 0 @. 0 @. 0@.\\ @. @VVV @VVV @VVV \\ 0 @»> F(M')... ...V @VVV \\ 0@»> F(I_{M'})@>\alpha» F(I_M)@>\beta » F(I_{M''})@»> 0 \end{CD}$$

such that each row in the last line is exact.

Note that $j$ -th cohomology of the complex $F(I_M)$ (respectively, $F(I_{M'}), F(I_{M''})$ ) gives the $R^jF(M) (respectively, (R^jF(M'), R^jF(M'')$ .) Using the resolution given in the lemma above, we may prove Theorem 7.1. Let us describe the map $\delta$ in more detail when $\mathcal{C}_2$ is a category of modules by ``diagram chasing''. Namely, for $x \in R^n (M'')$ , let us show how to compute $\delta(x)$ .

1. $x \in R^n (M'')$ may be represented as a class $[c_x]$ of a cocycle $c_x \in \operatorname{Ker}(d: F(I_{M''}^n) \to F(I_{M''}^{n+1}))$ .
2. We take a ``lift'' $\tilde c_x \in F(I^n_{M})$ such that $\beta^n(\tilde c_x)=c_x$ . Note that $\tilde c_x$ is no longer a cocycle.
3. Consider $e_x=d \tilde c_x \in F(I^{n+1} M)$ . It is a coboundary and we have $\beta(e_x)=0$ .
4. There thus exists an element $a_x \in F(I^n_{M'})$ such that $\alpha(a_x)= e_x$ . $a_x$ is no longer a coboundary but it is a cocycle.
5. The cohomology class $[a_x]$ of $a_x$ is the required $\delta(x)$ .

Such computation appears frequently when we deal with cohomologies.

DEFINITION 07.3   Let $A$ be a ring. Let $M,N$ be $A$ -modules. Then an extension of $N$ by $M$ is a module $L$ with a exact sequence

$$0 \to N \overset{\alpha}\to L \overset{\beta}\to M \to 0.$$ (E)

of $A$ -modules. Let

$$0 \to N \overset{\alpha'}\to L' \overset{\beta'}\to M \to 0.$$

be another extension. Then the two extensions are said to be isomorphic if there exists a commutative diagram

$$\begin{CD} 0 @»> N @>\alpha» L @>\beta» M @»> 0\\ @. @V=VV @VVV @V=VV @. \\ 0 @»> N @>\alpha'» L' @>\beta'» M @»> 0.\\ \end{CD}$$

PROPOSITION 07.4   There exists a bijection between the isomorphism classs of the extensions and elements of the $\operatorname{Ext}_A^1(M,N)$ . The bijection is given by corresponding the extension ($E$ ) to the class $\delta (1_N )\in \operatorname{Ext}^1(M,N)$ of the identity map $1_N$ by $\delta$ associated to the exact sequence $(E)$ .

See [1, XX,Exercise 27]