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+1dn = 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 }.KeywordsAbelian GroupExact SequenceRiemann SurfaceChain ComplexCohomology GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.