Homotopy types of truncated projective resolutions

TL;DR

This paper investigates the homotopy types of truncated projective resolutions over any ring R, demonstrating that such complexes can be stabilized to share the same homotopy type, revealing structural invariances.

Contribution

It introduces a method to stabilize truncated projective resolutions to achieve homotopy equivalence, extending understanding of their structural properties over arbitrary rings.

Findings

Truncated projective resolutions can be stabilized to have the same homotopy type.

The stabilization process applies over any ring R.

Homotopy types of these complexes are invariant under stabilization.

Abstract

We work over an arbitrary ring R. Given two truncated projective resolutions of equal length for the same module we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.