A criterion for sequential Cohen-Macaulayness

The purpose of this note is to show that a finitely generated graded module M over S=k[x1,& mldr;,xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=k[x_1,\ldots ,x_n]$$\end{document}, k a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree adeg(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {adeg}}(M)$$\end{document} agrees with adeg(F/ginrevlex(U))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))$$\end{document}, where F is a graded free S-module and M congruent to F/U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \cong F/U$$\end{document}. This answers positively a conjecture of Lu and Yu from 2016.