Topic
Alexander duality
About: Alexander duality is a research topic. Over the lifetime, 155 publications have been published within this topic receiving 5193 citations.
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12 Mar 2004
TL;DR: In this article, the authors propose a topology for the Geometry of Singular Spaces (GSPs) based on Generahties on Sheaves and derived tensor products.
Abstract: 1 Derived Categories.- 1.1 Categories of Complexes C*(A).- 1.2 Homotopical Categories K*(A).- 1.3 The Derived Categories D*(A).- 1.4 The Derived Functors of Hom.- 2 Derived Categories in Topology.- 2.1 Generahties on Sheaves.- 2.2 Derived Tensor Products.- 2.3 Direct and Inverse Images.- 2.4 The Adjunction Triangle.- 2.5 Local Systems.- 3 Poincare-Verdier Duality.- 3.1 Cohomological Dimension of Rings and Spaces.- 3.2 The Functor f!.- 3.3 Poincare and Alexander Duality.- 3.4 Vanishing Results.- 4 Constructible Sheaves, Vanishing Cycles and Characteristic Varieties.- 4.1 Constructible Sheaves.- 4.2 Nearby and Vanishing Cycles.- 4.3 Characteristic Varieties and Characteristic Cycles.- 5 Perverse Sheaves.- 5.1 t-Structures and the Definition of Perverse.- 5.2 Properties of Perverse.- 5.3 D-Modules and Perverse.- 5.4 Intersection Cohomology.- 6 Applications to the Geometry of Singular Spaces.- Singularities, Milnor Fibers and Monodromy.- Topology of Deformations.- Topology of Polynomial Functions.- Hyperplane and Hypersurface Arrangements.- References.
596 citations
TL;DR: In this paper, the authors give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear, and give various bounds for the projective dimension of such ideals, generalizing other recent results.
Abstract: In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.
211 citations
TL;DR: In this article, it was shown that the i-linear strand of the minimal free resolution of a Stanley-Reisner ideal IΔ⊂ S has the same information as the module structure of ExtiS(k[Δ∨,], εS), where Δ∨ is the Alexander dual of Δ.
194 citations
TL;DR: This paper states that reverse lexicographic initial ideals of generic lattice ideals are generic, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex.
92 citations
TL;DR: In this paper, it was shown that if a Legendrian knot in standard contact possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family.
Abstract: We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\Rightarrow$ normal ruling.
68 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 4 |
| 2019 | 5 |
| 2018 | 6 |
| 2017 | 7 |
| 2016 | 7 |