A criterion for sequential Cohen-Macaulayness

Abstract The purpose of this note is to show that a finitely generated graded module M over S=kx₁, , xₙ S = k x 1, …, x n, k a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree {adeg} (M) adeg (M) agrees with {adeg} (F/ {gin}ᵣevlex (U) ) adeg (F / gin revlex (U) ), where F is a graded free S -module and M F/U M ≅ F / U. This answers positively a conjecture of Lu and Yu from 2016.