publication . Article . Preprint . 2009

Residues and duality for singularity categories of isolated Gorenstein singularities

Daniel Murfet;
Open Access
  • Published: 08 Dec 2009 Journal: Compositio Mathematica, volume 149, pages 2,071-2,100 (issn: 0010-437X, eissn: 1570-5846, Copyright policy)
  • Publisher: Wiley
Abstract
We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.
Subjects
arXiv: Mathematics::Commutative AlgebraMathematics::Category Theory
free text keywords: Algebra and Number Theory, Mathematics - Commutative Algebra, Mathematics - Algebraic Geometry, Serre duality, Pairing, Pure mathematics, Duality (optimization), Injective function, Singularity theory, Algebra, Singularity, Mathematics, Gravitational singularity
23 references, page 1 of 2

M. R. Douglas, D-Branes, Categories, and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818- 2843, arXiv:hep-th/0011017.

T. Dyckerhoff and D. Murfet, The Kapustin-Li formula revisited, arXiv:1004.0687.

T. Dyckerhoff and D. Murfet, Pushing forward matrix factorisations, arXiv:1102.2957.

D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. [OpenAIRE]

E.E. Enochs and O.M.G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no.4, 611-633. [OpenAIRE]

P. Gabriel, Objets injectifs dans les cat´egories ab´eliennes, S´eminaire P. Dubriel (12e ann´ee 1958/1959), 17-01 to 17-32.

GLS07 G. M. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007.

J. Lipman and P. Sastry, Regular differentials and equidimensional scheme-maps, J. Alg. Geom. 1 (1992), 101-130.

MSV09 M. Mackaay, M. Stoˇsi´c, and P. Vaz, sl(N )-link homology (N > 4) using foams and the Kapustin-Li formula, Geom. Topol. 13 (2009), no. 2, 1075-1128.

, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.

D.G. Northcott, Injective envelopes and inverse polynomials, J. London Math. Soc. (2), 8 (1974), 290-296.

Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240-262.

P. Sastry and A. Yekutieli, On residue complexes, dualizing sheaves and local cohomology modules, Israel J. Math. 90 (1995), no. 1-3, 325-348. [OpenAIRE]

E. Segal, The closed state space of affine Landau-Ginzburg B-models, arXiv:0904.1339v1.

J.-P. Serre, Groupes Alg´ebriques et Corps de Classes, Hermann, Paris (1959).

23 references, page 1 of 2
Any information missing or wrong?Report an Issue