Im di−1A are called i-boundaries

1. Complexes

A (cochain) complex of modules over a ring R is a Z-graded R-module A =L

iAⁱ to- gether with a sequence of R-homomorphisms dⁱ_A : Aⁱ → Aⁱ⁺¹ such that dⁱ_A◦ dⁱ⁻¹_A = 0 We call A the underlying R-module and dⁱ_A the coboundary operators or differentials.

Convention: From now on, a complex of modules over the ring R is simply called a complex.

Let A be a complex. Elements of Zⁱ(A) = ker dⁱ_A are called i-cocycles while elements of Bⁱ(A) = Im dⁱ⁻¹_A are called i-boundaries. Since dⁱ_A◦ dⁱ⁻¹_A Im dⁱ⁻¹_A is contained in ker dⁱ_Aas an R-submodule. The quotient R-module Hⁱ(A) = Zⁱ(A)/Bⁱ(A) is called the i-th cohomology of complex A. Elements of Hⁱ(A) are called cohomology classes.

A morphism f : A → B of complexes consists of a sequence of R-homomorphisms fⁱ : Aⁱ → Bⁱ such that

dⁱ_B◦ fⁱ = fⁱ⁺¹◦ dⁱ_A for all i, i.e. we have the following commuting diagrams

(1.1)

A^{i d}

i

−−−−→ AA ⁱ⁺¹

fⁱ



 y



 y^f

i−1

B_i ^∂

0

−−−−→ Bi ⁱ⁺¹. Complexes over R forms a category C(R − mod).

Proposition 1.1. A morphism f : A → B of complexes sends cocycles to cocycles and coboundaries to coboundaries. Hence we obtain a well-defined homomorphism

Hⁱ(f ) : Hⁱ(A) → Hⁱ(B)

defined by Hⁱ(f )(z) = fⁱ(z). Here z means the cohomology class of z.

Proof. Exercise. 

Definition 1.1. Let f : A → B and g : A → B be morphisms of complexes. A homotopy from f to g is a sequence of R-homomorphisms hⁱ: Aⁱ→ Bⁱ⁻¹such that

fⁱ− gⁱ= dⁱ⁻¹_B ◦ hⁱ+ hⁱ⁺¹◦ dⁱ_A for all i.

Proposition 1.2. Homotopic maps induce equal maps in cohomology.

Proof. Exercise. 

We can also define the notion of chain complexes and homology. We will not define them here.

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