A criterion for sequential Cohen-Macaulayness

Abstract

The purpose of this note is to show that a finitely generated graded module M over $$S=k[x_1,\ldots ,x_n]$$ S = k [ x 1 , … , x n ] , k a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree $${\text {adeg}}(M)$$ adeg ( M ) agrees with $${\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))$$ adeg ( F / gin revlex ( U ) ) , where F is a graded free S-module and $$M \cong F/U$$ M ≅ F / U . This answers positively a conjecture of Lu and Yu from 2016.