Basic Notions of Algebra pp 213-230 | Cite as

# Homological Algebra

## Abstract

The algebraic aspect of homology theory is not complicated. A *chain complex* is a sequence {*C* _{ n }}_{ n∈ℤ} of Abelian groups (most often *C* _{ n } = 0 for *n* < 0) and connecting homomorphisms ∂_{ n }: *C* _{ n } → *C* _{ n−1}, called *boundary maps*; a *cochain complex* is a sequence {*C* ^{ n }}_{ n ∈ ℤ} of Abelian groups and homomorphisms *d* _{ n }: *C* ^{ n } → *C* ^{ n+1}, called *coboundary maps* or *differentials*. The boundary homomorphisms of a chain complex must satisfy the condition ∂_{ n }∂_{ n+1} = 0 for all *n* ∈ ℤ, and the co-boundary of a cochain complex the condition *d* _{ n+1}d_{n} = 0. Thus a complex is defined not just by the system of groups, but also by the homomorphisms, and we will for example denote a chain complex by *K* = {*C* _{ n }, ∂_{ n }}.

## Keywords

Abelian Group Exact Sequence Riemann Surface Chain Complex Cohomology Group## Preview

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