CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Derived categories We refer to [1] for a good guide to the theory.

Main idea: Instead of dealing with an object of an additive category 𝒞, we deal with complexes of 𝒞. But:

(1) We want to regard quasi-isomorphic complexes as the “same”.

(2) We want to identify two morphisms to be the same if they are homotopic.

11.1. Cone of a complex. Assume we are talking about complexes of objects in an additive category 𝒞.

Definition 11.1. [1, 4.1] For any complex X^{•}, we define T X^{•} to be a complex defined by

(T X)^{i} = X^{i+1}, d_{T X} = −d_{X}.

Definition 11.2. [1, 4.3] Let u : X^{•} → Y^{•} be a morphism of complexes. The cone C_{u}^{•} of u is defined to be a graded object

Y^{•} ⊕ T X^{•}

equipped with the following differential:

d (\begin{matrix}y \\ x \end{matrix}) = (\begin{matrix}d_{Y} & u \\ 0 & −d_{X} \end{matrix}) (\begin{matrix}y \\ x \end{matrix})

Idea 1: Instead of considering kernel and cokernel of a morphism u, we consider its cone C_{u}.

For any u, we have morphisms (triangle):

X^{•} →^{u} Y^{•} →^{ι_{Y}} C_{u}^{•} →^{p_{T X}} T X^{•}.

Let us call such a triangle standard. Now if 𝒞 is abelian, then for each standard triangle as above we have the following long exact sequence:

⋯ → H^{k}(X^{•}) → H^{k}(Y^{•}) → H^{k}(C_{u}^{•}) → H^{k+1}(X^{•}) → …

11.2. The category K(𝒞).

Definition 11.3. [1, 5.1] For any additive category 𝒞, we define K(𝒞) to be

(1) Ob(K(𝒞)) = Ob(C(𝒞)) (that means, objects of K(𝒞) are complexes).

(2) For any objects X^{•}, Y^{•} of K(𝒞), we define

Hom_{K(𝒞)}(X^{•}, Y^{•}) = Hom_{C(𝒞)}(X^{•}, Y^{•})/Homotopy

Even if 𝒞 is abelian, K(𝒞) is no longer abelian in general [1, 5.7]. But K(𝒞) has distinguished triangles, which are triangles isomorphic to standard triangles.

YOSHIFUMI TSUCHIMOTO

11.3. The cateogory D(𝒞). We assume 𝒞 is an abelian category. We then add some inverses of quasi isomorphisms in K(𝒞) to define D(𝒞). D(𝒞) again is not necessarily be an abelian category, but it is a triangulated category which has distinguished triangles which satisfy certain axioms.

By considering only complexes which are bounded below, we may define C^{+}(𝒞), K^{+}(𝒞), D^{+}(𝒞) etc.

Proposition 11.4. [1, 4.8] If 𝒞 has enoufh injectives then D^{+}(𝒞) is equivalent to K^{+}(I(𝒞)), where I(𝒞) is the category of injective objects in 𝒞.

So, in a sence, to consider an object X^{•} of D^{+}(𝒞) is to consider an injective resolution I^{•} of X^{•} and treat it up to homotopy.

For left-exact functor 𝒞_{1} → 𝒞_{2}, we may “define” (the actual definiton should be done more carefully. See [1])

ℝF : D^{+}(𝒞_{1}) → D^{+}(𝒞_{2})

by

ℝF (X^{•}) = F (I^{•})

where I^{•} is an injective resolution of X^{•}.

A good thing about treating derived functors in this way is that we may easily treat derived functors of compositions:

ℝ(F ∘ G) ≅ (ℝF) ∘ (ℝG).

References

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