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Chain Contraction

Let A be a commutative ring, let C_r be an R-module for r=0, 1, 2, ..., and define a chain complex C__ of the form

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

A chain contraction Gamma:0=1:C__->C__ is a collection of R-modules morphisms Gamma:C_r->C_(r+1) such that, for all r>=0,

dGamma+Gammad=1:C_r->C_r.

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

See also

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

This entry contributed by Christopher Stover

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References

Ranicki, A. "Notes on Reidemeister Torsion." 1997. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf.

Referenced on Wolfram|Alpha

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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