Syzygy Theorems via Comparison of Order Ideals on a Hypersurface

TL;DR

This paper introduces a new weak order ideal property to prove the Evans-Griffith Syzygy Theorem, comparing homological algebra over local rings and hypersurfaces, and solves key cases of the syzygy conjecture in mixed characteristic settings.

Contribution

It establishes a novel weak order ideal property for syzygy theorems and applies it to resolve important cases of the Evans-Griffith conjecture over certain local rings.

Findings

Proves the Evans-Griffith Syzygy Theorem under new conditions.

Solves specific cases of the syzygy conjecture over unramified mixed characteristic rings.

Reduces remaining problems to modules with finite projective dimension over hypersurfaces.

Abstract

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring $R$ versus a hypersurface ring $R/(x^n)$. Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic $p$, with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by $p^s$, $s>0$, that have finite projective dimension over a hypersurface ring.