Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T04:50:25.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Residues and duality for singularity categories of isolated Gorenstein singularities

Part of: Abelian categories Theory of modules and ideals

Published online by Cambridge University Press:  09 September 2013

Daniel Murfet
Daniel Murfet*
Affiliation:
UCLA, Los Angeles, CA 90095-1555, USA email daniel.murfet@math.ucla.edu
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

singularity categorymaximal Cohen–Macaulay modules

MSC classification

Secondary: 13C14: Cohen-Macaulay modules 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013 , pp. 2071 - 2100
Copyright
© The Author(s) 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auslander, M., Functors and morphisms determined by objects, in Representation theory of algebras, Proceedings, Conference at Temple University, Philadelphia, PA, 1976, Lecture Notes in Pure and Applied Mathematics, vol. 37 (Dekker, New York, 1978), 1–244.Google Scholar
Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. Lond. Math. Soc. (3) 85 (2002), 393–440.Google Scholar
Avramov, L. L. and Veliche, O., Stable cohomology over local rings, Adv. Math. 213 (2007), 93–139.Google Scholar
Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and reconstructions, Isz. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183–1205.Google Scholar
Brown, R., The twisted Eilenberg–Zilber theorem, Celebrazioni Archimedee del secolo XX, Simposio di topologia (Oderisi, 1967).Google Scholar
Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993), 34–37.Google Scholar
Buchweitz, R.-O., Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, Manuscript (1986), University of Hannover, available from: https://tspace.library.utoronto.ca/handle/1807/16682.Google Scholar
Buchweitz, R.-O., Greuel, G.-M. and Schreyer, F.-O., Cohen–Macaulay modules on hypersurface singularities II, Invent. Math. 88 (1987), 165–182.Google Scholar
Burban, I. and Drozd, Y., Maximal Cohen–Macaulay modules over surface singularities, in Trends in representation theory of algebras and related topics, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2008), 101–166.Google Scholar
Carqueville, N., Matrix factorisations and open topological string theory, J. High Energy Phys. 07 (2009), article 005; doi:10.1088/1126-6708/2009/07/005.Google Scholar
Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, NJ, 1956).Google Scholar
Christensen, L. W., Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747 (Springer, Berlin, 2000).Google Scholar
Conrad, B., Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750 (Springer, Berlin, 2000).Google Scholar
Crainic, M., On the perturbation lemma, and deformations, Preprint (2004), arXiv:math/0403266v1.Google Scholar
de Salas, F.-S., Residue: a geometric construction, Canad. Math. Bull. 45 (2002), 284–293.Google Scholar
Douglas, M. R., D-branes, categories, and $N= 1$ supersymmetry, J. Math. Phys. 42 (2001), 2818–2843; arXiv:hep-th/0011017.CrossRefGoogle Scholar
Dyckerhoff, T. and Murfet, D., Pushing forward matrix factorizations, Duke Math. J. 162 (2013), 1249–1311.Google Scholar
Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.Google Scholar
Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.Google Scholar
Gabriel, P., Objets injectifs dans les catégories abéliennes, Séminaire Dubreil. Algèbre et théorie des nombres 12 (1958/1959), Exp. No. 17, 32 p.Google Scholar
Greuel, G. M., Lossen, C. and Shustin, E., Introduction to singularities and deformations, Springer Monographs in Mathematics (Springer, Berlin, 2007).Google Scholar
Griffiths, P. and Harris, J., Principles of algebraic geometry, Pure and Applied Mathematics (Wiley, New York, 1978).Google Scholar
Gugenheim, V. K. A. M., On the chain complex of a fibration, Illinois J. Math. 16 (1972), 398–414.Google Scholar
Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).Google Scholar
Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, vol. 156 (Springer, Berlin, 1970).Google Scholar
Herbst, M. and Lazaroiu, C.-I., Localization and traces in open-closed topological Landau–Ginzburg models, J. High Energy Phys. 05 (2005), article 044.Google Scholar
Hübl, R. and Kunz, E., Integration of differential forms on schemes, J. Reine Angew. Math. 410 (1990), 53–83.Google Scholar
Jørgensen, P., Existence of Gorenstein projective resolutions and Tate cohomology, J. Eur. Math. Soc. (JEMS) 9 (2007), 59–76.Google Scholar
Kapustin, A. and Li, Y., D-branes in Landau–Ginzburg models and algebraic geometry, J. High Energy Phys. 12 (2003), article 005.Google Scholar
Kapustin, A. and Li, Y., Topological correlators in Landau–Ginzburg models with boundaries, Adv. Theor. Math. Phys. 7 (2003), 727–749.Google Scholar
Keller, B., Calabi–Yau triangulated categories, in Trends in representation theory of algebras, ed. Skowronski, A. (European Mathematical Society, Zurich, 2008).Google Scholar
Knörrer, H., Cohen–Macaulay modules on hypersurface singularities I, Invent. Math. 88 (1987), 153–164.Google Scholar
Kontsevich, M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians (Birkhäuser, Basel, 1995), 120–139; arXiv:alg-geom/9411018.Google Scholar
Krause, H., The stable derived category of a Noetherian scheme, Compositio Math. 141 (2005), 1128–1162.Google Scholar
Kunz, E., Residues and duality for projective algebraic varieties, University Lecture Series, vol. 47 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Lazaroiu, C. I., On the boundary coupling of topological Landau–Ginzburg models, J. High Energy Phys. 05 (2005), article 037.Google Scholar
Lipman, J., Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117 (1984).Google Scholar
Lipman, J., Residues and traces of differential forms via Hochschild homology, Contemporary Mathematics, vol. 61 (American Mathematical Society, Providence, RI, 1987).Google Scholar
Lipman, J., Lectures on local cohomology and duality, in Local cohomology and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 226 (Marcel Dekker, New York, 2001).Google Scholar
Lipman, J., Nayak, S. and Sastry, P., Pseudofunctorial behavior of Cousin complexes on formal schemes, in Variance and duality for cousin complexes on formal schemes, Contemporary Mathematics, vol. 375 (American Mathematical Society, Providence, RI, 2005), 3–132.Google Scholar
Mackaay, M., Stošić, M. and Vaz, P., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Mackaay, M., Stošić, M. and Vaz, P., $\mathfrak{s}\mathfrak{l}(N)$-link homology $(N\geq 4)$ using foams and the Kapustin–Li formula, Geom. Topol. 13 (2009), 1075–1128.Google Scholar
Northcott, D. G., Injective envelopes and inverse polynomials, J. Lond. Math. Soc. (2) 8 (1974), 290–296.Google Scholar
Orlov, D., Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), 240–262.Google Scholar
Sastry, P. and Yekutieli, A., On residue complexes, dualizing sheaves and local cohomology modules, Israel J. Math. 90 (1995), 325–348.Google Scholar
Segal, E., The closed state space of affine Landau–Ginzburg B-models, Preprint (2009), arXiv:0904.1339v1.Google Scholar
Serre, J.-P., Groupes Algébriques et Corps de Classes (Hermann, Paris, 1959).Google Scholar
Sharp, R. Y. and Zakeri, H., Modules of generalized fractions, Mathematika 29 (1982), 32–41.Google Scholar
Sharp, R. Y. and Zakeri, H., Local cohomology and modules of generalized fractions, Mathematika 29 (1982), 296–306.Google Scholar
Shih, W., Homology des espaces fibrés, Inst. Hautes Études Sci. 13 (1962), 93–176.Google Scholar
Tate, J., Residues of differentials on curves, Ann. Sci. Éc. Norm. Supér. (4) 1 (1968), 149–159.Google Scholar
Verdier, J.-L., Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), With a preface by L. Illusie, Edited and with a note by G. Maltsiniotis.Google Scholar
Witten, E., Mirror manifolds and topological field theory, in Essays on Mirror Manifolds, ed. Yau, S.-T. (International Press, 1992).Google Scholar
Witten, E., Chern–Simons gauge theory as a string theory, Preprint IASSNS-HEP-92/45 and hep-th/9207094.Google Scholar
Yoshino, Y., Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146 (Cambridge University Press, Cambridge, 1990).Google Scholar