CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Resolutions and derived functors

Definition 6.1. Let F : 𝒞_{1} → 𝒞_{2} be a functor between additive categories. We call F additive if for any objects M, N in 𝒞_{1},

Hom(M, N) → Hom(F (M), F (N))

is additive.

Definition 6.2. Let F be an additive functor from an abelian category 𝒞_{1} to 𝒞_{2}.

(1) F is said to be left exact (respectively, right exact ) if for any exact sequence

0 → L → M → N → 0,

the corresponding map

0 → F (L) → F (M) → F (N)

(respectively,

F (L) → F (M) → F (N) → 0)

is exact

(2) F is said to be exact if it is both left exact and right exact.

Definition 6.3. Let (K^{•}, d_{K}), (L^{•}, d_{L}) be complexes of objects of an additive category 𝒞.

(1) A morphism of complex u : K^{•} → L^{•} is a family

u^{j} : K^{j} → L^{j}

of morphisms in 𝒞 such that u commutes with d. That means,

u^{j+1} ∘ d_{K}^{j} = d_{K}^{j} ∘ u^{j}

holds.

(2) A homotopy between two morphisms u, v : K^{•} → L• of complexes is a family of morphisms

h^{j} : K^{j} → L^{j−1}

such that u − v = d ∘ h + h ∘ d holds.

Lemma 6.4. Let 𝒞 be an abelian category that has enough injectives. Then:

(1) For any object M in 𝒞, there exists an injective resolution of M. That means, there exists an complex I^{•} and a morphism ι_{M} : M → I^{0} such that

H^{j}(I^{•}) = \begin{cases}M (via ι_{M}) & if j = 0 \\ 0 & if j ≠ 0 \end{cases}

YOSHIFUMI TSUCHIMOTO

(2) For any morphism f : M → N of 𝒞, and for any injective resolutions (I^{•}, ι_{M}), (J^{•}, ι_{N}) of M and N (respectively), There exists a morphism f̄ : I^{•} → J^{•} of complexes which commutes with f. Forthermore, if there are two such morphisms f̄ and f′, then the two are homotopic.

Definition 6.5. Let 𝒞_{1} be an abelian category which has enough injectives. Let F : 𝒞_{1} → 𝒞_{2} be a left exact functor to an abelian category. Then for any object M of 𝒞_{1} we take an injective resolution I_{M}^{•} of M and define

R^{i}F (M) = H^{i}(I_{M}^{•}).

and call it the derived functor of F.

Lemma 6.6. The derived functor is indeed a functor.