A criterion for sequential Cohen-Macaulayness

Abstract

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

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