Homological criteria for minimal multiplicity

TL;DR

This paper provides new homological criteria, using Ext-algebras and linearity defect, to identify local rings with minimal multiplicity, enhancing understanding of their structure and properties.

Contribution

It introduces novel characterizations of minimal multiplicity rings through Ext-algebra properties and the linearity defect invariant.

Findings

Minimal multiplicity rings have Ext-algebras that are Gorenstein or Koszul AS-regular.

Linearity defect characterizations can detect minimal multiplicity.

Answers to Herzog and Iyengar's questions are provided in specific cases.

Abstract

Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar.