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Acyclic Chain Complex

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

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

is said to be acyclic if its rth homology group H_r(C__) is trivial for all values r>=0.

A straightforward result in homological algebra states that a chain complex C__ with each C_r free is acyclic if and only if there exists a chain contraction Gamma:0=1:C->C.

See also

Chain Complex, Chain Contraction, Commutative Ring, Free Module, Homological Algebra, Homology, Homology Group, Module

This entry contributed by Christopher Stover

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References

Ranicki, A. "Notes on Reidemeister Torsion." 1997. http://www.maths.ed.ac.uk/~aar/papers/torsion.pdf.

Cite this as:

Stover, Christopher. "Acyclic Chain Complex." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AcyclicChainComplex.html

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