Abstract
A chain complex is a sequence of abelian groups and homomorphisms
with the property d n ∘ d n+1 = 0 for all n. Homomorphisms d n are called boundary operators or differentials. A cochain complex is a sequence of abelian groups and homomorphisms
with the property d n ∘ d n−1 = 0. A chain complex can be considered as a cochain complex by reversing the enumeration: C n = C −n , d n = d −n . This is why we will usually consider only cochain complexes. A complex of A- modules is a complex for which C n (respectively C n) are modules over a ring A and d n (resp. d n) are homomorphisms of modules.
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© 1994 Springer-Verlag Berlin Heidelberg
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Kostrikin, A.I., Shafarevich, I.R. (1994). Complexes and Cohomology. In: Kostrikin, A.I., Shafarevich, I.R. (eds) Algebra V. Encyclopaedia of Mathematical Sciences, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57911-0_2
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DOI: https://doi.org/10.1007/978-3-642-57911-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65378-3
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