Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 06.1   Let $F: \mathcal{C}_1 \to \mathcal{C}_2$ be a functor between additive categories. We call $F$ additive if for any objects $M, N$ in $\mathcal{C}_1$ ,

$$\operatorname{Hom}(M,N)\to \operatorname{Hom}(F(M),F(N))$$

is additive.

DEFINITION 06.2   Let $F$ be an additive functor from an abelian category $\mathcal{C}_1$ to $\mathcal{C}_2$ .

1. $F$ is said to be left exact (respectively, right exact ) if for any exact sequence
   $$0 \to L\to M\to N\to 0,$$
   the corresponding map
   $$0\to F(L)\to F(M)\to F(N)$$
   (respectively,
   $$F(L)\to F(M)\to F(N) \to 0)$$
   is exact
2. $F$ is said to be exact if it is both left exact and right exact.

DEFINITION 06.3   Let $(K^\bullet,d_K)$ , $(L^\bullet, d_L)$ be complexes of objects of an additive category $\mathcal{C}$ .

1. A morphism of complex $u:K^\bullet \to L^\bullet$ is a family
   $$u^j: K^j \to L^j$$
   of morphisms in $\mathcal{C}$ such that $u$ commutes with $d$ . That means,
   $$u^{j+1} \circ d^j_K = d^j_K \circ u^j$$
   holds.
2. A homotopy between two morphisms $u,v: K^\bullet \to L\bullet$ of complexes is a family of morphisms
   $$h^j: K^j \to L^{j-1}$$
   such that $u-v = d \circ h + h \circ d$ holds.

LEMMA 06.4   Let $\mathcal{C}$ be an abelian category that has enough injectives. Then:

1. For any object $M$ in $\mathcal{C}$ , there exists an injective resolution of $M$ . That means, there exists an complex $I^\bullet$ and a morphism $\iota_M:M \to I^0$ such that
   $$% latex2html id marker 716H^j(I^\bullet) = \begin{cases} M... ...iota_M) &\text{ if } j=0 \\ 0 &\text{ if } j\neq 0 \end{cases}$$

2. For any morphism $f:M\to N$ of $\mathcal{C}$ , and for any injective resolutions $(I^\bullet,\iota_M)$ , $(J^\bullet,\iota_N)$ of $M$ and $N$ (respectively), There exists a morphism $\bar f:I^\bullet \to J^\bullet$ of complexes which commutes with $f$ . Forthermore, if there are two such morphisms $\bar f$ and $f'$ , then the two are homotopic.

DEFINITION 06.5   Let $\mathcal{C}_1$ be an abelian category which has enough injectives. Let $F: \mathcal{C}_1 \to \mathcal{C}_2$ be a left exact functor to an abelian category. Then for any object $M$ of $\mathcal{C}_1$ we take an injective resolution $I^\bullet_M$ of $M$ and define

$$R^i F(M)=H^i(I^\bullet_M).$$

and call it the derived functor of $F$ .

LEMMA 06.6   The derived functor is indeed a functor.