Homological Algebra
Tilting preserves finite global dimension
Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 563-570.

Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.72
Classification: 18G80, 18G20
Keller, Bernhard 1; Krause, Henning 2

1 Université de Paris, UFR de Mathématiques, Institut de Mathématiques de Jussieu–PRG, UMR 7586 du CNRS, Case 7012, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
2 Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
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Keller, Bernhard; Krause, Henning. Tilting preserves finite global dimension. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 563-570. doi : 10.5802/crmath.72. http://archive.numdam.org/articles/10.5802/crmath.72/

[1] Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning Handbook of tilting theory, London Mathematical Society Lecture Note Series, 332, Cambridge University Press, 2007 | MR | Zbl

[2] Baer, Dagmar Tilting sheaves in representation theory of algebras, Manuscr. Math., Volume 60 (1988) no. 3, pp. 323-347 | DOI | MR | Zbl

[3] Beĭlinson, Alexander A. Coherent sheaves on P n and problems in linear algebra, Funkts. Anal. Prilozh., Volume 12 (1978) no. 3, pp. 68-69 | MR

[4] Beĭlinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Société Mathématique de France, 1982, pp. 5-171 | Numdam | MR | Zbl

[5] Deligne, Pierre Cohomologie Etale. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4) (Dold, A.; Eckmann, B., eds.) (Lecture Notes in Mathematics), Volume 305, Springer, 1973 (Avec la collaboration de J.-F. Boutot, A. Grothendieck, L. Illusie et J.-L. Verdier)

[6] Gabriel, Peter Des catégories abéliennes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 323-448 | DOI | Numdam | Zbl

[7] Gabriel, Peter; Roĭter, Andrei V. Representations of finite-dimensional algebras, Encyclopaedia of Mathematical Sciences, 73, Springer, 1992, pp. 1-177 (With a chapter by B. Keller) | MR

[8] Geigle, Werner; Lenzing, Helmut A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Lecture Notes in Mathematics), Volume 1273, Springer, 1987, pp. 265-297 | DOI | MR | Zbl

[9] Happel, Dieter Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, 1988 | MR | Zbl

[10] Happel, Dieter Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, Proceedings, Paris 1987–1988 (39ème Année) (Malliavin, Marie-Paule, ed.) (Lecture Notes in Mathematics), Volume 1404, Springer (1989), pp. 108-126 | DOI | MR | Zbl

[11] Kapranov, Mikhail M. On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math., Volume 92 (1988) no. 3, pp. 479-508 | DOI | MR | Zbl

[12] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer, 2006 | MR | Zbl

[13] Keller, Bernhard Deriving DG categories, Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 1, pp. 63-102 | DOI | Numdam | MR | Zbl

[14] Keller, Bernhard Derived categories and their uses, Handbook of algebra. Vol. 1, Volume 1, North-Holland, 1996, pp. 671-701 | DOI | MR | Zbl

[15] Keller, Bernhard Hochschild cohomology and derived Picard groups, J. Pure Appl. Algebra, Volume 190 (2004) no. 1-3, pp. 177-196 | DOI | MR | Zbl

[16] Keller, Bernhard; Nicolás, Pedro Weight structures and simple dg modules for positive dg algebras, Int. Math. Res. Not., Volume 2013 (2013) no. 5, pp. 1028-1078 | DOI | MR | Zbl

[17] Keller, Bernhard; Vossieck, Dieter Sous les catégories dérivées, C. R. Math. Acad. Sci. Paris, Volume 305 (1987) no. 6, pp. 225-228 | Zbl

[18] Krause, Henning The stable derived category of a Noetherian scheme, Compos. Math., Volume 141 (2005) no. 5, pp. 1128-1162 | DOI | MR | Zbl

[19] Lenzing, Helmut Hereditary categories, Handbook of tilting theory (Hügel, Lidia Angeleri, ed.) (London Mathematical Society Lecture Note Series), Volume 332, Cambridge University Press, 2007, pp. 105-146 | DOI | MR

[20] Nagata, Masayoshi Local rings, Interscience Tracts in Pure and Applied Mathematics, 13, Interscience Publishers; John Wiley & Sons, 1962 | MR | Zbl

[21] Rickard, Jeremy Morita theory for derived categories, J. Lond. Math. Soc., Volume 39 (1989) no. 3, pp. 436-456 | DOI | MR | Zbl

[22] Rickard, Jeremy Derived equivalences as derived functors, J. Lond. Math. Soc., Volume 43 (1991) no. 1, pp. 37-48 | DOI | MR | Zbl

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