COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

03. cohomology of a complex. We mainly follow the treatment in [1].

Definition 3.1. Let R be a ring. A cochain complex of Rmodules is a sequence of R-modules

C^{•} : … →^{d^{n−1}} C^{n} →^{d^{n}} C^{n+1} →^{d^{n+1}} …

such that d^{n}∘d^{n−1} = 0 . The n-th cohomology of the cochain complex is defined to be the R-module

H^{n}(C^{•}) = Ker(d^{n})/ Image(d^{n−1}).

Elements of Ker(d^{n}) (respectively, Image(d^{n−1})) are often referred to as cocycles (respectively, coboundaries).

A bit of category theory:

Definition 3.2. A category 𝒞 is a collection of the following data (1) A collection Ob(𝒞) of objects of 𝒞.

(2) For each pair of objects X, Y ∈ Ob(𝒞), a set

Hom_{𝒞}(X, Y)

of morphisms.

(3) For each triple of objects X, Y, Z ∈ Ob(𝒞), a map(“composition (rule)”)

Hom_{𝒞}(X, Y) × Hom_{𝒞}(Y, Z) → Hom_{𝒞}(X, Z)

satisfying the following axioms (1) Hom(X, Y) ∩ Hom(Z, W) = ∅ unless (X, Y) = (Z, W).

(2) (Existence of an identity) For any X ∈ Ob(𝒞), there exists an element id_{X} ∈ Hom(X, X) such that

id_{X} ∘f = f, g ∘ id_{X} = g

holds for any f ∈ Hom(S, X), g ∈ Hom(X, T) (∀S, T ∈ Ob(𝒞)).

(3) (Associativity) For any objects X, Y, Z, W ∈ Ob(𝒞), and for any morphisms f ∈ Hom(X, Y), g ∈ Hom(Y, Z), h ∈ Hom(Z, W), we have

(f ∘ g) ∘ h = f ∘ (g ∘ h).

Morphisms are the basic actor/actoress in category theory.

An additive category is a category in which one may “add” some morphisms.

Definition 3.3. An additive category 𝒞 is said to be abelian if it satisfies the following axioms. (A4-1) Every morphism f : X → Y in 𝒞 has a kernel ker(f) : Ker(f) →

X. (A4-2) Every morphism f : X → Y in 𝒞 has a cokernel coker(f) : Y →

Coker(f).

YOSHIFUMI TSUCHIMOTO

(A4-3) For any given morphism f : X → Y, we have a suitably defined isomorphism

l : Coker(ker(f)) ≅ Ker(coker(f))

in 𝒞. More precisely, l is a morphism which is defined by the following relations:

ker(coker(f)) ∘ \overline{f} = f (∃\overline{f}), \overline{f} = l ∘ coker(ker(f)).

Definition 3.4. Let 𝒞 be an abelian category.

(1) An object I in 𝒞 is said to be injective if it satisfies the following condition: For any morphism f : M → I and for any monic morphism ι : N → M, f “extends” to a morphism f̂: M → I.

\begin{matrix}M & ⟶^{f̂} & I \\ ^{ι}↑ & & ∥ \\ N & ⟶^{f} & I \end{matrix}

(2) An object P in 𝒞 is said to be projective if it satisfies the following condition: For any morphism f : P → N and for any epic morphism π : M → N, f “lifts” to a morphism f̂: M → I.

\begin{matrix}P & ⟶^{f̂} & M \\ ∥ & & ^{π}↓ \\ P & ⟶^{f} & N \end{matrix}

Exercise 3.1. Let R be a ring. Let

0 → M_{1} → M_{2} → M_{3} → 0

be an exact sequence of R-modules. Assume furthermore that M_{3} is projective. Then show that the sequence

0 → Hom_{R}(N, M_{1}) →^{Hom_{R}(N,f)} Hom_{R}(N, M_{2}) →^{Hom_{R}(N,g)} Hom_{R}(N, M_{3}) → 0

is exact.

References

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