Acyclic Chain Complex

Let  be a commutative ring and let  be an R-module for . A chain complex  of the form

is said to be acyclic if its th homology group  is trivial for all values .

A straightforward result in homological algebra states that a chain complex  with each  free is acyclic if and only if there exists a chain contraction .

REFERENCES:

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