Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T11:16:52.487Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on thick subcategories of stable derived categories

Published online by Cambridge University Press:  11 January 2016

Henning Krause  and
Greg Stevenson
Henning Krause
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany, [email protected]
Greg Stevenson
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

Keywords

18E3013D0914F05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 212 , December 2013 , pp. 87 - 96
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories, J. Algebra 236 (2001), 819–834. MR 1813503. DOI 10.1006/jabr.2000.8529.CrossRefGoogle Scholar
[2] Beligiannis, A. and Krause, H., Thick subcategories and virtually Gorenstein algebras, Illinois J. Math. 52 (2008), 551–562. MR 2524651.CrossRefGoogle Scholar
[3] Benson, D. J., Carlson, J. F., and Rickard, J., Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59–80. MR 1450996.CrossRefGoogle Scholar
[4] Benson, D. J., Iyengar, S. B., and Krause, H., Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), 1643–1684. MR 2846489. DOI 10.4007/ annals.2011.174.3.6.CrossRefGoogle Scholar
[5] Bondal, A. and Bergh, M. van den, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1–36, 258. MR 1996800.CrossRefGoogle Scholar
[6] Buchweitz, R.-O., Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings, preprint, 1987.Google Scholar
[7] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. MR 0463157.Google Scholar
[8] Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), 1128–1162. MR 2157133. DOI 10.1112/S0010437X05001375.CrossRefGoogle Scholar
[9] Neeman, A., The derived category of an exact category, J. Algebra 135 (1990), 388– 394. MR 1080854. DOI 10.1016/0021-8693(90)90296-Z.CrossRefGoogle Scholar
[10] Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547–566. MR 1191736.CrossRefGoogle Scholar
[11] Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236. MR 1308405. DOI 10.1090/ S0894-0347-96-00174-9. ˇCrossRefGoogle Scholar
[12] Oppermann, S. and Št’ov´íček, J., Generating the bounded derived category and perfect ghosts, Bull. Lond. Math. Soc. 44 (2012), 285–298. MR 2914607. DOI 10.1112/blms/ bdr093.CrossRefGoogle Scholar
[13] Orlov, D. O., Triangulated categories of singularities and D-branes in Landau-Ginzburg models (in Russian), Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 240–262; English translation in Proc. Steklov Inst. Math. 246 (2004), 227–248. MR 2101296.Google Scholar
[14] Quillen, D., “Higher algebraic K-theory, I” in Algebraic K-Theory, I: Higher K- Theories (Seattle, 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973, 85–147. MR 0338129.Google Scholar
[15] Schoutens, H., Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217–239. MR 1969220. DOI 10.1081/AGB-120016756.CrossRefGoogle Scholar
[16] Stevenson, G., Subcategories of singularity categories via tensor actions, preprint, arXiv:1105.4698v3 [math.AG]CrossRefGoogle Scholar
[17] Takahashi, R., Classifying thick subcategories of the stable category of Cohen- Macaulay modules, Adv. Math. 225 (2010), 2076–2116. MR 2680200. DOI 10.1016/ j.aim.2010.04.009.CrossRefGoogle Scholar
[18] Takahashi, R., Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the punctured spectra, J. Math. Soc. Japan 65 (2013), 357–374.CrossRefGoogle Scholar
[19] Verdier, J.-L., Des catégories dérivées des catégories abéliennes, Astérisque 239, Soc. Math. France, Paris, 1997. MR 1453167.Google Scholar