70 research outputs found
Intersections of Leray complexes and regularity of monomial ideals
For a simplicial complex X and a field K, let h_i(X)=\dim \tilde{H}_i(X;K).
It is shown that if X,Y are complexes on the same vertex set, then for all k
h_{k-1}(X\cap Y) \leq \sum_{\sigma \in Y} \sum_{i+j=k} h_{i-1}(X[\sigma])\cdot
h_{j-1}(\lk(Y,\sigma)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0
for all induced subcomplexes Y \subset X and i \geq d. Let L_K(X) denote the
minimal d such that X is d-Leray over K. The above theorem implies that if X,Y
are simplicial complexes on the same vertex set then L_K(X \cap Y) \leq L_K(X)
+L_K(Y). Reformulating this inequality in commutative algebra terms, we obtain
the following result conjectured by Terai: If I,J are square-free monomial
ideals in S=K[x_1,...,x_n], then reg(I+J) \leq reg(I)+reg(J)-1 where reg(I)
denotes the Castelnuovo-Mumford regularity of I.Comment: 9 page
Higher chordality: From graphs to complexes
We generalize the fundamental graph-theoretic notion of chordality for higher
dimensional simplicial complexes by putting it into a proper context within
homology theory. We generalize some of the classical results of graph
chordality to this generality, including the fundamental relation to the Leray
property and chordality theorems of Dirac.Comment: 13 pages, revised; to appear in Proc. AM
Matchings, coverings, and Castelnuovo-Mumford regularity
We show that the co-chordal cover number of a graph G gives an upper bound
for the Castelnuovo-Mumford regularity of the associated edge ideal. Several
known combinatorial upper bounds of regularity for edge ideals are then easy
consequences of covering results from graph theory, and we derive new upper
bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio
Poset topology and homological invariants of algebras arising in algebraic combinatorics
We present a beautiful interplay between combinatorial topology and
homological algebra for a class of monoids that arise naturally in algebraic
combinatorics. We explore several applications of this interplay. For instance,
we provide a new interpretation of the Leray number of a clique complex in
terms of non-commutative algebra.
R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie
combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent
naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs
applications de cette interaction. Par exemple, nous introduisons une nouvelle
interpr\'etation du nombre de Leray d'un complexe de clique en termes de la
dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159
and an article in preparation. 12 pages, 3 Figure
Regularity of squarefree monomial ideals
We survey a number of recent studies of the Castelnuovo-Mumford regularity of
squarefree monomial ideals. Our focus is on bounds and exact values for the
regularity in terms of combinatorial data from associated simplicial complexes
and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
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