A criterion for sequential Cohen-Macaulayness

TL;DR

This paper provides a new criterion for determining when a finitely generated graded module over a polynomial ring is sequentially Cohen-Macaulay, based on the equality of arithmetic degrees involving generic initial modules.

Contribution

It proves a conjecture by Lu and Yu (2016) that characterizes sequential Cohen-Macaulay modules via arithmetic degrees and generic initial modules.

Findings

Characterization of sequential Cohen-Macaulay modules using arithmetic degrees

Proof of Lu and Yu's conjecture from 2016

Equivalence involving generic initial modules and arithmetic degrees

Abstract

The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm gin}_{revlex}(U))$, where $F$ is a graded free $S$-module and $M \cong F/U$. This answers positively a conjecture of Lu and Yu from 2016.