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183
Resolutions Of Stanley-Reisner Rings And Alexander Duality
, 1996
"... Associated to any simplicial complex \Delta on n vertices is a square-free monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the Stanley-Reisner ring. This note considers a simplicial complex which is in a sense a canonical ..."
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Cited by 120 (2 self)
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Associated to any simplicial complex \Delta on n vertices is a square-free monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the Stanley-Reisner ring. This note considers a simplicial complex which is in a sense a canonical Alexander dual to \Delta, previously considered in [Ba, BrHe]. Using Alexander duality and a result of Hochster computing the Betti numbers dim k Tor i (k[\Delta]; k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in \Delta . As corollaries, we prove that I \Delta has a linear resolution as A-module if and only if \Delta is Cohen-Macaulay over k, and show how to compute the Betti numbers dim k Tor i (k[\Delta]; k) in some cases where \Delta is well-behaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.
The LCM-lattice in monomial resolutions
, 1999
"... Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: ..."
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Cited by 61 (7 self)
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Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were:
On the radical of a monomial ideal
- J. ALGEBRA
, 2004
"... Algebraic and combinatorial properties of a monomial ideal and its radical are compared. ..."
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Cited by 61 (9 self)
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Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
On the Charney-Davis and Neggers-Stanley Conjectures
"... For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the Neggers-Stanley conjecture on real zeroes for W (P, t) in t ..."
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Cited by 43 (6 self)
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For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the Neggers-Stanley conjecture on real zeroes for W (P, t) in these cases. The result is deduced from McMullen's g-Theorem, by exhibiting a simplicial polytopal sphere whose h-polynomial is W (P, t). Whenever this...
Finite filtrations of modules and shellable multicomplexes
- MANUSCRIPTA MATH
, 2005
"... We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes. ..."
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Cited by 39 (7 self)
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We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.
Simplicial trees are sequentially Cohen-Macaulay
- J. Pure Appl. Algebra
"... This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a sim-plicial tree is a componentwise linear ..."
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Cited by 38 (8 self)
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This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a sim-plicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties of simplicial trees. The main result of the this paper is that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. Sequentially Cohen-Macaulay modules were introduced by Stanley [S] (following the introduction of nonpure shellability by Bjorner and Wachs [BW]) so that a nonpure shellable simplicial complex had a sequentially Cohen-Macaulay Stanley-Reisner ideal. Herzog and Hibi ([HH]) then dened the notion of a componentwise linear ideal, which extended a criterion of Eagon and Reiner ([ER]) for Cohen-Macaulayness of an ideal to a criterion for sequential Cohen-Macaulayness. Simplicial trees, on the other hand, were introduced in [F1] in the context of Rees rings, and their facet ideals were studied further in [F2] for their Cohen-Macaulay properties, and
SHELLABLE GRAPHS AND SEQUENTIALLY COHEN-MACAULAY 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 non-pure sense of Björner-Wachs. 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 non-pure sense of Björner-Wachs. 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 Cohen-Macaulay bipartite graphs. We also give an inductive procedure to build all such shellable bipartite graphs. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.
Discrete Morse Theory for Cellular Resolutions
- J. Reine Angew. Math
, 2000
"... We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for gen ..."
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Cited by 34 (2 self)
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We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for generic and shellable monomial modules. The latter ones are introduced in this paper and include stable, squarefree stable monomial ideals and Stanley-Reisner ideals I such that is the combinatorial Alexander-dual of a non-pure shellable simplicial complex. For generic monomial ideals the given resolution is the resolution determined by the Scarf-complex as shown by Bayer & Sturmfels and Bayer, Peeva & Sturmfels and by Miller, Sturmfels & Yanagawa in its most general form. For stable and squarefree-stable monomial ideals the minimal resolution had been determined by Eliahou & Kervaire respectively Aramova, Herzog & Hibi and Peeva by algebraic means. In great detail we examplify our method for powers of the maximal ideal. Parallel to our work Skoldberg has constructed minimal free resolutions for a class of ideals that includes the shellable ones by algebraic means. The preceding constructions can be viewed as a generalization of a result by Lyubeznik which gives a resolution that is a subcomplex of the Taylor resolution. For ane semigroup rings k[] we are able to give a minimal free resolution of the eld k in case all lower intervals in the poset are non-pure shellable. This is a condition which implies in the pure case, in other words k[] is homogeneous, that k[] is Koszul. 1.
