CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Categories of modules. We first review definition and some basic properties of modules.

Definition 4.1. Let R be a (unital associative) ring. A set M is said to be an R-module if there is given a binary map (“action”)

R × M ∋ (r, m) ↦ r.m ∈ M

such that the following properties hold.

(1) ∀r ∈ R∀m_{1}, ∀m_{2} ∈ M r.(m_{1} + m_{2}) = r.m_{1} + r.m_{2}.

(2) ∀r_{1}, ∀r_{2} ∈ R∀m ∈ M (r_{1} + r_{2}).m = r_{1}.m + r_{2}.m.

(3) ∀m ∈ M 1_{R}.m = m.

(4) ∀r_{1}, ∀r_{2} ∈ R∀m ∈ M (r_{1}r_{2}).m = r_{1}.(r_{2}.m)

Definition 4.2. Let R be a (unital associative) ring. A map φ from an R-module M to another R-module N is an R-module homomorphism if the following conditions are satisfied.

(1) φ is additive. That means, we have

∀m_{1}∀m_{2} ∈ M φ(m_{1} + m_{2}) = φ(m_{1}) + φ(m_{2}).

(2) φ preserves the R-action. That means,

∀r ∈ R∀m ∈ M φ(r.m) = r.φ(m).

Proposition 4.3. For any given ring R, The category (R-mod) of R-modules is an abelian category.

Exercise 4.1. Let A be a 2 × 2 matrix

A = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})

over ℂ. We define a structure of ℂ[X]-module on V = ℂ^{2} by putting

f(X).v = f(A)v (v ∈ ℂ^{2})

Then:

(1) Show that V has a proper ℂ[X]-submodule W. (That means, ℂ[X] submodule such that W ≠ V, 0.

(2) Show that there is no other proper submodule of V.

Definition 4.4. A cochain complex in an abelian category 𝒞 is a sequence of objects and morphisms in 𝒞

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

such that d^{n} ∘ d^{n−1} = 0.

Cohomology objects of the cochain complex are

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