The purpose of this note is to show that a finitely generated graded module M over S=k[x₁,… ,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/ginᵣevlex(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.
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Caviglia et al. (2024) studied this question.