A criterion for sequential Cohen-Macaulayness

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