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Dimension filtration, sequential Cohen-Macaulayness and a new polynomial invariant of graded algebras
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2016 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 456, 250-265 p.Article in journal (Refereed) Published
Abstract [en]

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 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.

Place, publisher, year, edition, pages
Academic Press, 2016. Vol. 456, 250-265 p.
Keyword [en]
Sequential Cohen-Macaulayness, Hilbert series, Initial ideal, Extremal Betti numbers
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186125DOI: 10.1016/j.jalgebra.2016.01.045ISI: 000375237300011ScopusID: 2-s2.0-84960890577OAI: oai:DiVA.org:kth-186125DiVA: diva2:925571
Note

QC 20160509

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2016-05-30Bibliographically approved
In thesis
1. Topological and Shifting Theoretic Methods in Combinatorics and Algebra
Open this publication in new window or tab >>Topological and Shifting Theoretic Methods in Combinatorics and Algebra
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, 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 generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2016. 152 p.
Series
TRITA-MAT-A, 2016:02
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186136 (URN)978-91-7595-899-6 (ISBN)
Public defence
2016-06-07, F3, Lindstedtsvägen 26, Stockholm, 12:30 (English)
Opponent
Supervisors
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-02 Last updated: 2016-05-16Bibliographically approved

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