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Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).ORCID-id: 0000-0002-7497-2764
Freie Universität, Germany.
2017 (Engelska)Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, nr 12, s. 3851-3865Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

Ort, förlag, år, upplaga, sidor
European Mathematical Society Publishing House, 2017. Vol. 19, nr 12, s. 3851-3865
Nyckelord [en]
Simplicial complex, face numbers, Stanley-Reisner rings, sequential Cohen-Macaulayness, componentwise linear ideals
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:kth:diva-186130DOI: 10.4171/JEMS/755ISI: 000415853100009Scopus ID: 2-s2.0-85035041682OAI: oai:DiVA.org:kth-186130DiVA, id: diva2:925575
Forskningsfinansiär
Vetenskapsrådet, 2011-11677-88409-18
Anmärkning

QC 20171207

Tillgänglig från: 2016-05-02 Skapad: 2016-05-02 Senast uppdaterad: 2017-12-07Bibliografiskt granskad
Ingår i avhandling
1. Topological and Shifting Theoretic Methods in Combinatorics and Algebra
Öppna denna publikation i ny flik eller fönster >>Topological and Shifting Theoretic Methods in Combinatorics and Algebra
2016 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
KTH Royal Institute of Technology, 2016. s. 152
Serie
TRITA-MAT-A ; 2016:02
Nationell ämneskategori
Matematik
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kth:diva-186136 (URN)978-91-7595-899-6 (ISBN)
Disputation
2016-06-07, F3, Lindstedtsvägen 26, Stockholm, 12:30 (Engelska)
Opponent
Handledare
Anmärkning

QC 20160516

Tillgänglig från: 2016-05-16 Skapad: 2016-05-02 Senast uppdaterad: 2016-05-16Bibliografiskt granskad

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