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

2015/04/16 by Afshin Goodarzi, Goodarzi, Afshin · 1 citation
Mathematics · #05E40 #05E45 #13D45 #13P10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:05E40 #msc:05E45 #msc:13D45 #msc:13P10

paper · pdf · doi:10.48550/arxiv.1504.04328

arxiv created 2015/04/16 · arxiv updated 2015/04/17

Abstract

Let \k be a field and let A be a standard ℕ-graded \k-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), that we call the Björner--Wachs polynomial, to A. It is shown that the Björner--Wachs polynomial is an algebraic counterpart of 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 connections with the Hilbert series of local cohomology modules.

Cited by

• On sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules

Related

• Sequentially Cohen--Macaulayness of bigraded modules
• Sequentially Cohen-Macaulay modules and local cohomology
• Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties
• Sequentially Cohen-Macaulay monomial ideals of embedding dimension four
• On the structure of sequentially Cohen--Macaulay bigraded modules