Modules of infinite regularity over commutative graded rings

TL;DR

This paper investigates the regularity properties of modules over non-Koszul graded commutative algebras, establishing conditions under which certain modules have infinite Castelnuovo-Mumford regularity and relating these properties to algebraic deviations.

Contribution

It proves that non-Koszul graded algebras have modules with infinite regularity and connects the vanishing of graded deviations to module regularity over complete intersections.

Findings

Modules of the form mM have infinite regularity in non-Koszul algebras.

Nonzero summands of syzygies of k have infinite regularity over non-Koszul complete intersections.

Vanishing graded deviations relate to finite regularity of syzygy summands.

Abstract

In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal of $R$ and by $M$ a finitely generated graded $R$-module, the nonzero modules of the form $\mathfrak{m} M$ have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of $k$ has infinite regularity. Finally, we relate the vanishing of the graded deviations of $R$ to having a nonzero direct summand of a syzygy of $k$ of finite regularity.