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Reconstructing schemes from the derived category

Part of: (Co)homology theory Abelian categories

Published online by Cambridge University Press:  16 March 2012

Carlos Sancho de Salas  and
Fernando Sancho de Salas
Carlos Sancho de Salas
Affiliation:
Department of Mathematics, Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain (fsancho@usal.es; mplu@usal.es)
Fernando Sancho de Salas
Affiliation:
Department of Mathematics, Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain (fsancho@usal.es; mplu@usal.es)
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Abstract

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We generalize Bondal and Orlov's Reconstruction Theorem for a Gorenstein scheme X and a projective morphism X → T whose (relative) dualizing sheaf is either T-ample or T-antiample.

Keywords

derived categoriesequivalencesreconstructionlinear functors

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 781 - 796
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Avramov, L. L., Iyengar, S. B., and Lipman, J., Reflexivity and rigidity for complexes, II, Schemes, preprint (arXiv:1001.3450v1; math.AG).Google Scholar
2.Ballard, M., Derived categories of sheaves on singular schemes with an application to reconstruction, Adv. Math. 227 (2011), 895–919.CrossRefGoogle Scholar
3.Balmer, P., Presheaves of triangulated categories and reconstruction of schemes, Math. Annalen 324 (2002), 557–580.CrossRefGoogle Scholar
4.Balmer, P., The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168.CrossRefGoogle Scholar
5.Bartocci, C., Bruzzo, U., Ruipérez, D. Hernández and Porras, J. M. Muñoz, Mirror symmetry on K3 surfaces via Fourier–Mukai transform, Commun. Math. Phys. 195 (1998), 79–93.CrossRefGoogle Scholar
6.Bondal, A. I. and Orlov, D. O., Semi orthogonal decomposition for algebraic varieties, MPIM Preprint 95/15 (arXiv:alg-geom/9506012v1; 1995).Google Scholar
7.Bondal, A. I. and Orlov, D. O., Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), 327–344.CrossRefGoogle Scholar
8.Bridgeland, T., Fourier–Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133.CrossRefGoogle Scholar
9.Bridgeland, T., Flops and derived categories, Invent. Math. 147 (2002), 613–632.CrossRefGoogle Scholar
10.Bridgeland, T. and Maciocia, A., Fourier–Mukai transforms for K3 and elliptic fibrations, J. Alg. Geom. 11 (2002), 629–657.CrossRefGoogle Scholar
11.Chen, J.-C., Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Diff. Geom. 61 (2002), 227–261.Google Scholar
12.Ruipérez, D. Hernández, Martín, A. C. López and de Salas, F. Sancho, Fourier–Mukai transforms for Gorenstein schemes, Adv. Math. 211 (2007), 594–620.CrossRefGoogle Scholar
13.Hernández|Ruipérez, D., Martín, A. C. López and de Salas, F. Sancho, Relative integral functors for singular fibrations and singular partners, J. Eur. Math. Soc. 11 (2009), 597–625.CrossRefGoogle Scholar
14.Illusie, L., Existence de résolutions globals, in Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Exposé 2, Lecture Notes in Mathematics, Volume 225 (Springer, 1971).Google Scholar
15.Mukai, S., Duality between D(X) and D() with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175.CrossRefGoogle Scholar
16.Mukai, S., On the moduli space of bundles on K3 surfaces, I, in Vector bundles on algebraic varieties, Tata Institute of Fundamental Research Studies in Mathematics, Volume 11, pp. 341–413 (Tata Institute of Fundamental Research, Bombay, 1987).Google Scholar
17.Orlov, D. O., Equivalences of derived categories and K3 surfaces, J. Math. Sci. 84 (1997), 1361–1381.CrossRefGoogle Scholar
18.Polishchuk, A., Symplectic biextensions and a generalization of the Fourier–Mukai transform, Math. Res. Lett. 3 (1996), 813–828.CrossRefGoogle Scholar