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Chain Contraction


A chain contraction Gamma:0=1:C__->C__ of a chain complex C__ over a commutative ring R is a collection of R-module morphisms Gamma:C_r->C_(r+1) satisfying

 dGamma+Gammad=1:C_r->C_r

for all r>=0. Here, each C_r is an R-module, and the chain complex has the form

 C__:...->C_n->C_(n-1)->C_(n-2)->...->C_2->C_1->C_0.

The map d:C_r->C_(r-1) is the boundary operator of C__.


See also

Boundary Operator, Chain Complex, Commutative Ring, Module, Morphism, R-Module

Portions of this entry contributed by Christopher Stover

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References

Ranicki, A. "Notes on Reidemeister Torsion." 2001. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf.

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Chain Contraction

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Chain Contraction." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChainContraction.html

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