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38
SPLITTABLE IDEALS AND THE RESOLUTIONS OF MONOMIAL IDEALS
, 2006
"... We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic squarefree monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Pa ..."
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Cited by 37 (6 self)
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We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic squarefree monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary squarefree monomial ideals viewed as facet ideals of simplicial complexes.
SHELLABLE GRAPHS AND SEQUENTIALLY COHENMACAULAY BIPARTITE GRAPHS
, 2007
"... Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the nonpure sense of BjörnerWachs. We are then interested in determining what families of graphs have t ..."
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Cited by 36 (7 self)
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Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the nonpure sense of BjörnerWachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially CohenMacaulay bipartite graphs. We also give an inductive procedure to build all such shellable bipartite graphs. Because shellable implies that the associated StanleyReisner ring is sequentially CohenMacaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) CohenMacaulay. We also give a new proof for a result of Faridi on the sequentially CohenMacaulayness of simplicial forests.
Sequentially CohenMacaulay edge ideals
 PROC. AMER. MATH. SOC.
, 2007
"... Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x1,...,xn] denote its associated edge ideal. We show that all chordal graphs G are sequentially CohenMacaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Fari ..."
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Cited by 35 (4 self)
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Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x1,...,xn] denote its associated edge ideal. We show that all chordal graphs G are sequentially CohenMacaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi’s theorem that the facet ideal of a simplicial tree is sequentially CohenMacaulay and implies Herzog, Hibi, and Zheng’s theorem that a chordal graph is CohenMacaulay if and only if its edge ideal is unmixed. We also characterize the sequentially CohenMacaulay cycles and produce some examples of nonchordal sequentially CohenMacaulay graphs.
RESOLUTIONS OF SQUAREFREE MONOMIAL IDEALS VIA FACET IDEALS: A SURVEY
, 2006
"... We survey some recent results on the minimal graded free resolution of a squarefree monomial ideal. The theme uniting these results is the pointofview that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a ne ..."
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Cited by 22 (3 self)
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We survey some recent results on the minimal graded free resolution of a squarefree monomial ideal. The theme uniting these results is the pointofview that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the StanleyReisner correspondence, to associate to a squarefree monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current stateoftheart with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part
Whiskers and sequentially CohenMacaulay graphs
, 2006
"... Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and ..."
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Cited by 21 (6 self)
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Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G∪W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially CohenMacaulay graph. For instance, we show that if S is a vertex cover of G, then G ∪ W(S) is a sequentially CohenMacaulay graph. On the other hand, we show that if G\S is not sequentially CohenMacaulay, then G∪W(S) is not a sequentially CohenMacaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get CohenMacaulay graphs.
Monomial ideals via squarefree monomial ideals
 In Commutative algebra, volume 244 of Lect. Notes Pure Appl. Math
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Some families of componentwise linear monomial ideals
, 2006
"... Let R = k[x1,...,xn] be a polynomial ring over a field k. Let J = {j1,...,jt} be a subset of {1,...,n}, and let mJ ⊂ R denote the ideal (xj1,..., xjt). Given subsets J1,...,Js of {1,..., n} and positive integers a1,..., as, we study ideals of the form I = m a1 ..."
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Cited by 13 (3 self)
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Let R = k[x1,...,xn] be a polynomial ring over a field k. Let J = {j1,...,jt} be a subset of {1,...,n}, and let mJ ⊂ R denote the ideal (xj1,..., xjt). Given subsets J1,...,Js of {1,..., n} and positive integers a1,..., as, we study ideals of the form I = m a1