CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Resolutions and derived functors We recommend the book of Lang [1] as a good reference. The treat-

ment here follows the book for the most part.

Theorem 7.1. Let 𝒞_{1} be an abelian category with enough injectives, and let F : 𝒞_{1} → 𝒞_{2} be a covariant additive left functor to another abelian category 𝒞_{2}. Then:

(1) F ≅ R^{0}F.

(2) For each short exact sequence

0 → M′ → M → M′′ → 0

and for each n ≥ 0 there is a natural homomorphism

δ^{n} : R^{n}F (M′′) → R^{n+1}F (M)

such that we obtain a long exact sequence

⋯ → R^{n}F (M′) → R^{n}F (M) → R^{n}F (M′′) →^{δ^{n}} R^{n+1}F (M′) → ….

(3) δ is natural. That means, for a morphism of short exact se-

quences

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0
↓  ↓  ↓
0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0

the δ’s give a commutative diagram:

R^{n}F (M′′) ⟶^{δ^{n}} R^{n+1}F (M′)
↓  ↓
R^{n}F (N′′) ⟶^{δ^{n}} R^{n+1}F (N′)

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

Lemma 7.2. For any exact sequence 0 → M′ → M → M′′ → 0 of objects in 𝒞_{1}, There exists injective resolutions I_{M′}, I_{M}, I_{M′′} of M′, M, M′′ respectively and a commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0
↓  ↓  ↓
0 ⟶ I_{M′} ⟶ I_{M} ⟶ I_{M′′} ⟶ 0

such that the diagram of resolutions is exact.

Definition 7.3. 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.

YOSHIFUMI TSUCHIMOTO

Theorem 7.4. Let

0 → M → X^{0} → X^{1} → X^{2} → …

be a resolution of M by F -acyclics. Let

0 → M → I^{0} → I^{1} → I^{2} → …

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

H^{n}F (X) ≅ H^{n}(F (I)) = R^{n}F (M)

for all n ≥ 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.

References

[1] S. Lang, Algebra (graduate texts in mathematics), Springer Verlag, 2002.

[2] P.P.Grivel, Catégorie dérivées et foncteurs dérivés, In: Algebraic D-modules, Perspectives in mathematics 2 (1997), 1–108.