Cellular resolutions of monomial modules

Bayer, Dave; Sturmfels, Bernd

doi:10.1515/crll.1998.083

Cited by 160 publications

(252 citation statements)

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“…The most noteworthy such families include the Koszul complex, the Eagon-Northcott complex [17], and the resolution of generic monomial ideals [3] (see also [4]). In Section 3 we analyze even further the minimal free resolution of a Ferrers ideal I λ and obtain a surprisingly elegant description of the differentials in the resolution in Theorem 3.2.…”

Section: Figure 1 Ferrers Graph Tableau and Idealmentioning

confidence: 99%

“…In some sense, this is a prototypical result as it provides the minimal free resolution of several classes of ideals obtained from Ferrers ideals by appropriate specializations of the variables (see [13] for further details). Our description of the free resolution of a Ferrers ideal relies on the theory of cellular resolutions as developed by Bayer and Sturmfels in [3] (see also [41]). More precisely, let ∆ n−1 × ∆ m−1 denote the product of two simplices of dimensions n − 1 and m − 1, respectively.…”

Section: Figure 1 Ferrers Graph Tableau and Idealmentioning

confidence: 99%

“…For our construction we use cellular resolutions and polyhedral cell complexes as introduced by Bayer and Sturmfels in [3]. First, we briefly recall some basic notions, but we refer to [3] (or [41]) for a more detailed introduction to the topic. A polyhedral cell complex X is a finite collection of convex polytopes (in some R N ) called faces (or cells) of X such that:…”

Section: Minimal Free Resolutions Of Ferrers Idealsmentioning

confidence: 99%

“…However, this ideal can be obtained as a specialization of a suitable Ferrers ideal. 3 ] be a polynomial ring over a field K and let I be the edge ideal corresponding to a cycle of length three, that is, I = (x 1 x 2 , x 1 x 3 , x 2 x 3 ). Clearly, the ideal I does not arise from a bipartite graph.…”

Section: Theorem 42 Let G Be a Bipartite Graph Without Isolated Vermentioning

confidence: 99%

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Monomial and toric ideals associated to Ferrers graphs

Corso

Nagel

2008

Trans. Amer. Math. Soc.

106

121

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Abstract. Each partition λ = (λ 1 , λ 2 , . . . , λ n ) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

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Section: Figure 1 Ferrers Graph Tableau and Idealmentioning

confidence: 99%

Section: Figure 1 Ferrers Graph Tableau and Idealmentioning

confidence: 99%

Section: Minimal Free Resolutions Of Ferrers Idealsmentioning

confidence: 99%

Section: Theorem 42 Let G Be a Bipartite Graph Without Isolated Vermentioning

confidence: 99%

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Monomial and toric ideals associated to Ferrers graphs

Corso

Nagel

2008

Trans. Amer. Math. Soc.

106

121

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“…Let V b be the subcomplex of V induced by the vertices whose corresponding coatoms do not lie above e 1 …”

Section: Proof Under These Conditions Each (D−i)-cell Is In Exactly mentioning

confidence: 99%

Topological representations of matroids

Swartz

2002

J. Amer. Math. Soc.

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There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky’s enumeration of the cells of the arrangement still holds. Bounded subcomplexes of an arrangement of homotopy spheres correspond to minimal cellular resolutions of the dual matroid Steiner ideal. As a result, the Betti numbers of the ideal are computed and seen to be equivalent to Stanley’s formula in the special case of face ideals of independence complexes of matroids.

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The Alexander Duality Functors and Local Duality with Monomial Support

Miller

2000

Journal of Algebra

120

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Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated ‫ގ‬ n -graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every ‫ޚ‬ n -graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all ‫ގ‬ n -graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen᎐Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the n i Ž . ‫ޚ‬ -graded local cohomology functors H y for squarefree monomial ideals I in I the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I s ᒊ is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, i Ž . that Terai's formula for the Hilbert series of H S is equivalent to Hochster's for

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Cellular resolutions of monomial modules

Abstract: Abstract:We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS], [PS].

Cited by 160 publications

References 10 publications

Monomial and toric ideals associated to Ferrers graphs

Monomial and toric ideals associated to Ferrers graphs

Topological representations of matroids

The Alexander Duality Functors and Local Duality with Monomial Support

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