Homological Algebra on a Complete Intersection, with an Application to Group Representations

Eisenbud, David

doi:10.2307/1999875

Cited by 290 publications

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“…Localizing at a; b; c we can assume that S and R are local rings. Note that BB H B H B a n b n c n ´Id S 4 and all the matrix elements of B and B H belong to the maximal ideal of S. Thus B; B H is a reduced matrix factorization of a n b n c n over S (see [5]). In particular, by [5,Corollary 6.3], the complex…”

Section: Local Orbifold Euler Numbers For Ordinary Singularitiesmentioning

confidence: 99%

“…Since g Ã g Ã b S n E ÃÃ is re¯exive, we get the equivalence of (2) and (3). Obviously, (3) implies (4), and (4) implies (5). To prove that (3) implies (5) note that g Ã b S n E ÃÃ is an O Y e -submodule of n th symmetric power of meromorphic 1-forms on e Y generated by g Ã q for q P b S n E. Hence it is suf®cient to prove that the germ of g Ã q belongs to b S n F P; Y e for every point P P e Y.…”

Section: The Logarithmic Rami®cation Formula and A Vanishing Theorem mentioning

confidence: 99%

“…So it is suf®cient to prove that the natural map coker B 3 im A is an isomorphism. Since det B a n b n c n 2 and a n b n c n is a prime, coker B is a rank-2 re¯exive R-module by [5,Proposition 5.6]. Since AB 0 and both coker B and im A are rank-2 re¯exive modules, it is suf®cient to check that coker B is a direct summand of R 4 at every prime of R with height 1.…”

Section: Local Orbifold Euler Numbers For Ordinary Singularitiesmentioning

confidence: 99%

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Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Langer¹

2003

Proc. Lond. Math. Soc.

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We introduce orbifold Euler numbers for normal surfaces with boundary $\mathbb{Q}$-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov–Miyaoka–Yau type inequality. Existence of such a generalization was earlier conjectured by G. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282]. Most of the paper is devoted to properties of local orbifold Euler numbers and to their computation.As a first application we show that our results imply a generalized version of R. Holzapfel's ‘proportionality theorem’ [Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg, Braunschweig, 1998)]. Then we show a simple proof of a necessary condition for the logarithmic comparison theorem which recovers an earlier result by F. Calderón-Moreno, F. Castro-Jiménez, D. Mond and L. Narváez-Macarro [Comment. Math. Helv. 77 (2002) 24–38].Then we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type (conjectured by G. Tian [Springer Lecture Notes in Mathematics 1646 (1996) 143–185)]. Finally, we give some applications to singularities of plane curves; for example, we improve F. Hirzebruch's bound on the maximal number of cusps of a plane curve.2000 Mathematical Subject Classification: 14J17, 14J29, 14C17.

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Section: Local Orbifold Euler Numbers For Ordinary Singularitiesmentioning

confidence: 99%

Section: The Logarithmic Rami®cation Formula and A Vanishing Theorem mentioning

confidence: 99%

Section: Local Orbifold Euler Numbers For Ordinary Singularitiesmentioning

confidence: 99%

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Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Langer¹

2003

Proc. Lond. Math. Soc.

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“…for a single homogeneous relation h among the f i 's, there will always be such an R-free resolution of M which is eventually 2-periodic (see, for example, [35, § 6] or [10]).…”

Section: Springer-type Resultsmentioning

confidence: 99%

Extending the coinvariant theorems of Chevalley, Shephard-Todd, Mitchell, and Springer

Broer

Reiner

Smith³

et al. 2011

Proceedings of the London Mathematical Society

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“…While matrix factorizations may seem strange, in fact, they arise very naturally in homological algebra (see, for example, Eisenbud [4]). Consider a module N , and a ring element ' 2 S which annihilates N .…”

Section: Definition 1 a (‫-ޚ‬Graded) Matrix Factorization On M With Pmentioning

confidence: 99%

Khovanov–Rozansky homology via a canopolis formalism

Webster

2007

Algebr. Geom. Topol.

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In this paper, we describe a canopolis (ie categorified planar algebra) formalism for Khovanov and Rozansky's link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Khovanov-Rozansky homology and the definition using Soergel bimodules in a more general context, allow us to give a new proof of the invariance of triply graded homology and give a new analysis of the behavior of triply graded homology under the Reidemeister IIb move. 57M27; 13D02In [9; 10], Khovanov and Rozansky introduced a series of homology theories for links. These theories categorify the quantum invariants for sl n , and the HOMFLYPT polynomial. Unfortunately, they remain very difficult to calculate, not least because of the complicated matrix factorizations used in their original combinatorial definition. Later work of I Frenkel, Khovanov and Stroppel [6;8;14;15] has suggested a more systematic definition of these invariants and a connection between these theories and the structure of the BGG category O for the Lie algebra gl n , but progress toward computational simplifications along these lines has been slow.In this paper, we will show that these invariants can be understood, computed and in fact, defined in the context of canopolises. We hope that this approach will both lead to computational benefits and help the reader to understand the definition of KhovanovRozansky homology better. A canopolis 1 is a categorification of the notion of a planar algebra defined by Bar-Natan [2] (see Section 2).Consider a disk in the plane with m disks removed from its interior (we call the places left by these removed disks "holes"). An oriented planar arc diagram (or "spaghettiand-meatballs diagram") Á on this disk is a collection of oriented simple curves with endpoints on the boundary of the disk (including the boundary of the holes), along with choice of a distinguished point on each boundary of a component (in diagrams, this point is distinguished by putting a star next to it), and an ordering of the holes of the diagram.1 Bar-Natan uses the term "canopoly" in the published version of [2], but the consensus choice now seems to be the more etymologically correct "canopolis." Let Q i .Á/ be the set of planar arc diagrams ! such that the outer boundary of ! matches the boundary of the i -th hole of Á. That is, there is the same number of endpoints, and if we order the endpoints, starting at our distinguished point, the orientations of the arcs match.Any fixed planar arc diagram Á with a distinguished hole defines an operationby shrinking the given m-tuple of diagrams, and pasting it into the holes of Á, as is shown in Figure 1. This "multiplication" is a particular instance of an algebraic structure called an colored operad.The operad of planar arc diagrams acts on tangle diagrams in a disk as well. Phrased in the language of [2], the set of tangle diagrams T S; in a disk, with endpoints on the boundary an...

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Homological Algebra on a Complete Intersection, with an Application to Group Representations

Cited by 290 publications

References 0 publications

Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Extending the coinvariant theorems of Chevalley, Shephard-Todd, Mitchell, and Springer

Khovanov–Rozansky homology via a canopolis formalism

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