A-graded methods for monomial ideals

Berkesch, Christine; Matusevich, Laura Felicia

doi:10.1016/j.jalgebra.2009.07.007

Cited by 3 publications

(7 citation statements)

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“…To compute the rank of M , we then apply Euler-Koszul homology to this resolution and follow the resulting spectral sequence, as in the proof of [BM09,Theorem 4.5]. That argument and the fact that the theorem has been proven for A-hypergeometric systems in [Berk10,Theorem 7.3] imply that this procedure and its associated numerics are compatible with ρ, and this compatibility yields the desired result.…”

Section: Binomial D-modulesmentioning

confidence: 99%

Torus equivariant D-modules and hypergeometric systems

Berkesch

Matusevich

Walther

2019

Advances in Mathematics

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We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor ΠÃ B on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. By applying ΠÃ B to suitable binomial D-modules, we shed new light on the D-module theoretic properties of systems of classical hypergeometric differential equations.

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Section: Binomial D-modulesmentioning

confidence: 99%

Torus equivariant D-modules and hypergeometric systems

Berkesch

Matusevich

Walther

2019

Advances in Mathematics

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“…The program computes the reliability polynomials for these systems with 10 components and 10 levels of performance in less than a minute. Figure 4 shows the reliability polynomials of a k-out-of-10 decreasing MS system for which k 1 to k 10 are (9,8,7,6,5,5,4,4,3,2). The program took 40 seconds on a laptop 1 , hence the method is practical.…”

Section: 3mentioning

confidence: 99%

“…The choice between steps (3) or (3') depends on our needs. If we are only interested in computing the full reliability formula, then we can use any algorithm that computes Hilbert series in step (3). However, if we need bounds for our system reliability, then we can compute any free resolution of I S,j and thus perform step (3').…”

Section: 2mentioning

confidence: 99%

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Polarization and depolarization of monomial ideals with application to multi-state system reliability

et al. 2019

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Polarization is a powerful technique in algebra which provides combinatorial tools to study algebraic invariants of monomial ideals. We study the reverse of this process, depolarization which leads to a family of ideals which share many common features with the original ideal. Given a squarefree monomial ideal, we describe a combinatorial method to obtain all its depolarizations, and we highlight their similar properties such as graded Betti numbers. We show that even though they have many similar properties, their differences in dimension make them distinguishable in applications in system reliability theory. In particular, we apply polarization and depolarization tools to study the reliability of multi-state coherent systems via binary systems and vice versa. We use depolarization as a tool to reduce the dimension and the number of variables in coherent systems. arXiv:1807.05743v2 [math.AC]

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“…Berkesch and Matusevich [BeMa08] used an extension [DMM08] of these homological methods to more general multigraded modules over polynomial rings to see what the Cohen-Macaulay characterization by rank jumps says about quotients by monomial ideals. Their conclusions were based on the fact that ranks of systems of differential equations arising from monomial ideals are controlled by the geometry of their distractions, from which Berkesch and Matusevich defined the exponent simplicial complexes using standard pairs.…”

Section: Distracting Arrangements and Exponent Complexesmentioning

confidence: 99%

Topological Cohen-Macaulay criteria for monomial ideals

Miller¹

2009

Combinatorial Aspects of Commutative Algebra

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IntroductionScattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. It is unclear whether researchers thinking about this topic have, to this point, been aware of the full spectrum of related developments. Therefore, the purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.Four families of simplicial complexes are defined in reverse chronological order: via distraction, theČech complex, Alexander duality, and then the Koszul complex. Each comes with historical remarks and context, including forays into Stanley decompositions, standard pairs, A-hypergeometric systems, cellular resolutions, thě Cech hull, polarization, local duality, and duality for Z n -graded resolutions. Results or definitions appearing here for the first time include the general categorical definition of cellular complex in Definition 3.2, as well as the statements and proofs of Lemmas 3.9 and 3.10, though these are very easy. The characterization of exponent simplicial complexes in Corollary 2.9 might be considered new; certainly the consequent connection in Theorem 4.1 to theČech simplicial complexes is new, as is the duality in Theorem 6.3 between these and the dualČech simplicial complexes. The geometric connection between distraction and local cohomology in Theorem 4.7, and its consequences in Section 5, generalize and refine results from [BeMa08], in addition to providing commutative proofs. Finally, the connection betweenČech and Koszul simplicial complexes in Lemma 7.2 and Corollary 7.8, as well as its general Z n -graded version in Theorem 7.7, appear to be new.

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A-graded methods for monomial ideals

Cited by 3 publications

References 18 publications

Torus equivariant D-modules and hypergeometric systems

Torus equivariant D-modules and hypergeometric systems

Polarization and depolarization of monomial ideals with application to multi-state system reliability

Topological Cohen-Macaulay criteria for monomial ideals

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