Dimension filtration, sequential Cohen–Macaulayness and a new polynomial invariant of graded algebras

Let kk be a field and let A   be a standard NN-graded kk-algebra. Using numerical information of some invariants in the primary decomposition of 0 in A  , namely the so-called dimension filtration, we associate a bivariate polynomial BW(A;t,w)BW(A;t,w), that we call the Björner–Wachs polynomial, to A.It is shown that the Björner–Wachs polynomial is an algebraic counterpart to the combinatorially defined h-triangle of finite simplicial complexes introduced by Björner & Wachs. We provide a characterisation of sequentially Cohen–Macaulay algebras in terms of the effect of the reverse lexicographic generic initial ideal on the Björner–Wachs polynomial. More precisely, we show that a graded algebra is sequentially Cohen–Macaulay if and only if it has a stable Björner–Wachs polynomial under passing to the reverse lexicographic generic initial ideal. We conclude by discussing some connections with the Hilbert series of local cohomology modules, extremal Betti numbers and combinatorial Alexander duality.