Homotopy categories of unbounded complexes of projective modules

TL;DR

This paper develops a stable theory for unbounded complexes of projective modules, establishing a duality criterion for exactness over generically Gorenstein rings and exploring the implications for total reflexivity.

Contribution

It introduces a stable category framework for projective complexes and proves a duality criterion for exactness in this context, advancing the understanding of module properties over Gorenstein rings.

Findings

A stable theory for projective complexes is constructed.

A complex is exact iff its dual complex is exact over a generically Gorenstein ring.

The work links total reflexivity conditions to duality properties.

Abstract

We develop in this paper a stable theory for projective complexes, by which we mean to consider a chain complex of finitely generated projective modules as an object of the factor category of the homotopy category modulo split complexes. As a result of the stable theory we are able to prove that a complex of finitely generated projective modules over a generically Gorenstein ring is exact if and only if its dual complex is exact. This shows the dependence of total reflexivity conditions for modules over a generically Gorenstein ring.