cochain complex

cochain complex

[′kō‚chān ′käm‚pleks]
(mathematics)
A sequence of Abelian groups Cn,- ∞ <>n < ∞,="" together="" with="" coboundary="" homomorphisms=""> n : CnC n +1such that δ n +1- δ n = 0.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Chataur, Saralegi-Aranguren, and Tanreuse a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field is isomorphic to the intersection cohomology of a pseudo-manifold introduced by M.
If an N-graded ring also is a cochain complex, we come to the following notion [5,17].
An N-graded K-ring [[OMEGA].sup.*] is called the differential graded ring (henceforth, DGR) if it is a cochain complex of K-modules
The cochain complex (26) is called the de Rham complex of a DGR ([[OMEGA].sup.*], [delta]).
The homology of the cochain complex ([C.sup.*] (G,A), [[partial derivative].sup.*]) is called the cohomology of G with coefficients in A
[2, Definition 5.2] The subcomplex of invariants denoted C[S.sup.n](G,A) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the symmetric cochain complex. Its homology is the symmetric cohomology of G with coefficients in A and is denoted H[S.sup.n](G,A) = Z[S.sup.n](G,A)/B[S.sup.n](G,A).
A cochain complex is a Z-graded differential module E = [[direct sum].sub.i[member of]Z] [E.sup.i] whose differential d has degree 1, which means d : [E.sup.n] [right arrow] [E.sup.n+1].
is a cochain complex and d is an antiderivation, i.e.
where here [C.sub.op] for a cochain complex C denotes the opposite chain complex that one obtains by reindexing in the opposite order and reversing all the arrows.
Likewise the map U makes them into cochain complexes, i.e., [U.sup.2] = 0.
Let us define a map k: D(A) [right arrow] D(A') of cochain complexes by k{[sigma]) = 0 if [x.sub.0] [member of] [sigma] and k([sigma]) = a if [x.sub.0] [??] [sigma].