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122
MONOMIAL IDEALS, EDGE IDEALS OF HYPERGRAPHS, AND THEIR MINIMAL GRADED FREE RESOLUTIONS
"... We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). ..."
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Cited by 58 (5 self)
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We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work [21] on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properlyconnected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved”. For such a hypergraph H (and thus, for any simple graph), we give upper and lower bounds for the regularity of I(H) via combinatorial information describing H. We also introduce triangulated hypergraphs, a properlyconnected hypergraph which is a generalization of chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.
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 36 (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.
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 34 (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.
RESIDUE CURRENTS WITH PRESCRIBED ANNIHILATOR IDEALS
, 2007
"... Abstract. Given a coherent ideal sheaf J we construct locally a vectorvalued residue current R whose annihilator is precisely the given sheaf. In case J is a complete intersection, R is just the classical ColeffHerrera product. By means of these currents we can extend various results, previously k ..."
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Cited by 31 (11 self)
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Abstract. Given a coherent ideal sheaf J we construct locally a vectorvalued residue current R whose annihilator is precisely the given sheaf. In case J is a complete intersection, R is just the classical ColeffHerrera product. By means of these currents we can extend various results, previously known for a complete intersection, to general ideal sheaves. Combining with integral formulas we obtain a residue version of the EhrenpreisPalamodov fundamental principle. 1.
Four positive formulas for type A quiver polynomials
, 2006
"... We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe ..."
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Cited by 31 (4 self)
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We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rcgraphs). Three of our formulae are multiplicityfree and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torusinvariant scheme. The remaining (presently nongeometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled ” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82, Jos84, BB85, Ros89] via equivariant Chow groups [EG98]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds
Characterizations of border bases
 Journal of Pure and Applied Algebra
"... This paper presents characterizations of border bases of zerodimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of ..."
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Cited by 30 (4 self)
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This paper presents characterizations of border bases of zerodimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of Spolynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute. Key words: border basis, division algorithm, syzygy module 1
POSITROID VARIETIES I: JUGGLING AND GEOMETRY
, 2009
"... While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coi ..."
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Cited by 29 (4 self)
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While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, and BrownGoodearlYakimov. However, its cyclicinvariance is hidden in this description. Postnikov gave many cyclicinvariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We adopt his terminology and call the strata positroid varieties. We show that positroid varieties are normal and CohenMacaulay, and are defined as schemes by the vanishing of Plücker coordinates. We compute their Tequivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and BuchKreschTamvakis ’ approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the HodgeGröbner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective StanleyReisner scheme of a shellable ball.
Planar Graphs As Minimal Resolutions Of Trivariate Monomial Ideals
, 2001
"... We introduce a new kind of plane drawing for simple triconnected planar graphs, called rigid embeddings in grid surfaces. They provide methods to (1) solve the problem of nding wellstructured (cellular, in this case) minimal free resolutions for arbitrary monomial ideals in three variables; (2) ..."
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Cited by 23 (1 self)
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We introduce a new kind of plane drawing for simple triconnected planar graphs, called rigid embeddings in grid surfaces. They provide methods to (1) solve the problem of nding wellstructured (cellular, in this case) minimal free resolutions for arbitrary monomial ideals in three variables; (2) give a new proof of the BrightwellTrotter bound on the order dimension of triconnected planar maps; and (3) generalize Schnyder's angle coloring of planar triangulations to arbitrary triconnected planar maps. Rigid embedding is stable under duality for planar maps, and has certain uniqueness properties.
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.