Eventually linear partially complete resolutions over a local ring with $m^4=0$

TL;DR

This paper classifies the Hilbert polynomial of certain local rings with nilpotent maximal ideal and explores properties of their resolutions, revealing symmetry constraints and non-periodicity in Betti sequences.

Contribution

It provides a classification of Hilbert polynomials for local rings with $m^4=0$ admitting specific resolutions and establishes new symmetry and periodicity constraints.

Findings

Hilbert polynomial classification for $m^4=0$ local rings with special resolutions

Resolutions can only be asymmetric if the Hilbert polynomial is symmetric

Betti sequences cannot be periodic with period two or three

Abstract

We classify the Hilbert polynomial of a local ring $(R,m)$ satisfying $m^4=0$ which admits an eventually linear doubly-infinite resolution $C$ which is 'partially' complete --- that is, for which the cohomology of $Hom_R(C,R)$ eventually vanishes. As a corollary to our main result, we show that an $m^4=0$ local ring can admit certain classes of asymmetric partially complete resolutions only if its Hilbert polynomial is symmetric. Moreover, we show that the Betti sequence associated to an eventually linear partially complete resolution over an $m^4=0$ local ring cannot be periodic of period two or three.