Homological Algebra

  • Igor R. Shafarevich
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 11)


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 }.


Abelian Group Exact Sequence Riemann Surface Chain Complex Cohomology Group 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of ScienceMoscowRussia

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