Poset resolutions and lattice-linear monomial ideals

Clark, Timothy B. P.

doi:10.1016/j.jalgebra.2009.11.029

Cited by 16 publications

(24 citation statements)

References 11 publications

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“…(a) It is now straightforward to check that (1.2) holds also for n = 0. (b) It follows from the definitions that the chain complex q≥0 C q,• (P, k) is canonically isomorphic to the part in non-negative homological degrees of the shifted poset construction D • ( P , k)(1) from [9], where P is the ranked poset obtained from P by using the canonical procedure from [9, Proposition A.9]. Thus the conic chain complex C • (P, k) should be thought of as a refined and better behaved version of D • (P, k).…”

Section: The Conic Chain Complexmentioning

confidence: 99%

See 1 more Smart Citation

Minimal free resolutions of monomial ideals and of toric rings are supported on posets

Clark

Tchernev

2018

Trans. Amer. Math. Soc.

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We introduce the notion of a resolution supported on a poset. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution. Work of Velasco has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is instead a homology CW-poset that supports the minimal free resolution of the ideal. In general there is more than one choice for the isomorphism class of such a poset, and it is an open question whether there is a canonical one.

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Section: The Conic Chain Complexmentioning

confidence: 99%

“…In this paper, we show that chain complexes arising from posets have the necessary degree of generality. We introduce the notion of a resolution supported on a poset, which is a refined and better behaved version of the notion of poset resolution from [9]. When the poset is a CW-poset, i.e.…”

Section: Introductionmentioning

confidence: 99%

Minimal free resolutions of monomial ideals and of toric rings are supported on posets

Clark

Tchernev

2018

Trans. Amer. Math. Soc.

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“…Lcm-lattices, which were introduced by Gasharov, Peeva, and Welker [Gasharov et al 1999], have become an important tool used in studying free resolutions of monomial ideals. There have been a number of results that use the lcm-lattice to give constructive methods for finding free resolutions for monomial ideals; for some examples see [Clark 2010;Peeva and Velasco 2011;Velasco 2008].…”

Section: Intersection (Meet Semi)latticesmentioning

confidence: 99%

Partially ordered sets in Macaulay2

Cook

Mapes

Whieldon

2015

JSAG

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We introduce the package Posets for Macaulay2. This package provides a data structure and the necessary methods for working with partially ordered sets, also called posets. In particular, the package implements methods to enumerate many commonly studied classes of posets, perform operations on posets, and calculate various invariants associated to posets. INTRODUCTION.A partial order is a binary relation over a set P that is antisymmetric, reflexive, and transitive. A set P together with a partial order is called a poset, or partially ordered set. We refer the reader to the seminal text [Stanley 2012] for definitions omitted herein.Posets are combinatorial structures that are used in modern mathematical research, particularly in algebra. We introduce the package Posets for Macaulay2 [Grayson and Stillman] via three distinct posets or related ideals which arise naturally in combinatorial algebra.We first describe two posets that are generated from algebraic objects. The intersection semilattice associated to a hyperplane arrangement can be used to compute the number of unbounded and bounded real regions cut out by a hyperplane arrangement, as well as the dimensions of the homologies of the complex complement of a hyperplane arrangement.Given a monomial ideal, the lcm-lattice of its minimal generators gives information on the structure of the free resolution of the original ideal. Specifically, two monomial ideals with isomorphic lcm-lattices have the "same" (up to relabeling) minimal free resolution, and the lcm-lattice can be used to compute, among other things, the multigraded Betti numbers

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“…Cocellular resolutions in [11] are obtained by M -homogenizing a cochain complex. The construction is also used in [4], [13]. …”

Section: Frames and Degenerations Of Monomial Resolutions 2033mentioning

confidence: 99%

“…Special cases of this construction have been used by several authors: for example, this is how simplicial resolutions [2], cellular resolutions [3], the Buchsbaum-Rim resolutions [5], and the cocellular resolutions [11] are built; it is also used in [4], [13]. A strong advantage of our approach is that it does not need a special choice of basis in the resolution (that is, any frame can be used), while all the special cases listed above need it.…”

Section: Introductionmentioning

confidence: 99%