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Books on the topic 'Categorías abelianas'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Borceux, Francis. Mal'cev, protomodular, homological and semi-abelian categories. Dordrecht: Kluwer Academic, 2004.

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2

1942-, Reiten Idun, and Smalø Sverre O, eds. Tilting in Abelian categories and quasitilted algebras. Providence, R.I: American Mathematical Society, 1996.

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3

Borceux, Francis, and Dominique Bourn. Mal’cev, Protomodular, Homological and Semi-Abelian Categories. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-1962-3.

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4

Definable additive categories: Purity and model theory. Providence, R.I: American Mathematical Society, 2011.

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5

Borceux, Francis. Handbook of categorical algebra 2: Categories and structures. Cambridge [England]: Cambridge University Press, 1994.

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6

Bernstein, Joseph. Equivariant sheaves and functors. Berlin: Springer-Verlag, 1994.

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7

Verdier, Jean Louis. Des catégories dérivées des catégories abéliennes. Paris: Société mathématique de France, 1996.

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8

1943-, Schapira Pierre, ed. Ind-sheaves. Paris: Société mathématique de France, 2001.

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9

A non-Hausdorff completion: The Abelian category of C-complete left modules over a topological ring. New Jersey: World Scientific, 2015.

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10

Pantev, Tony. Stacks and catetories in geometry, topology, and algebra: CATS4 Conference Higher Categorical Structures and Their Interactions with Algebraic Geometry, Algebraic Topology and Algebra, July 2-7, 2012, CIRM, Luminy, France. Providence, Rhode Island: American Mathematical Society, 2015.

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11

Dimitric, Radoslav. Slenderness: Volume 1, Abelian Categories. Cambridge University Press, 2018.

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12

Schneiders, Jean-Pierre. Quasi-Abelian Categories and Sheaves. Societe Mathematique De France, 1999.

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13

Huybrechts, D. Triangulated Categories. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0001.

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Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.

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14

Borceux, Francis, and Dominique Bourn. Mal'Cev, Protomodular, Homological and Semi-abelian Categories. Springer-Verlag New York Inc., 2004.

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15

Borceux, Francis, and Dominique Bourn. Mal'cev, Protomodular, Homological and Semi-Abelian Categories. Springer, 2010.

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16

Huybrechts, D. Derived Categories of Surfaces. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0012.

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This chapter completes the classification of algebraic surfaces from the point of view of their derived categories. Abelian, K3, and elliptic surfaces play a special role. For all other surfaces, the derived category determines the isomorphism type. The reduction to minimal surfaces is due to Kawamata, and the case of elliptic surfaces was dealt with by Bridgeland and Maciocia.

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17

Huybrechts, D. Derived Categories: A Quick Tour. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0002.

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This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.

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18

Huybrechts, D. Derived Categories of Coherent Sheaves. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0003.

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The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are constructed. The usual geometric functors, direct and inverse image, tensor product, and global sections, are derived and extended to functors between derived categories. The compatibilities between them are reviewed. The final section focuses on the Grothendieck-Verdier duality.

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19

Bakalov, Bojko, and Alexander Kirillov. Lectures on Tensor Categories and Modular Functors (University Lecture Series). American Mathematical Society, 2000.

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20

The Beilinson Complex And Canonical Rings of Irregular Surfaces (Memoirs of the American Mathematical Society). American Mathematical Society, 2006.

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21

Variance And Duality For Cousin Complexes On Formal Schemes (Contemporary Mathematics). American Mathematical Society, 2005.

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22

Huybrechts, D. K3 Surfaces. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0010.

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After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

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23

Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001.

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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

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