A criterion for sequential Cohen-Macaulayness

Abstract

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