A criterion for sequential Cohen-Macaulayness

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