Skip to main content
×
×
Home

A note on thick subcategories of stable derived categories

  • Henning Krause (a1) and Greg Stevenson (a2)
Abstract
Abstract

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.

Copyright
References
Hide All
[1] Balmer P. and Schlichting M., Idempotent completion of triangulated categories, J. Algebra 236 (2001), 819834. MR 1813503. DOI 10.1006/jabr.2000.8529.
[2] Beligiannis A. and Krause H., Thick subcategories and virtually Gorenstein algebras, Illinois J. Math. 52 (2008), 551562. MR 2524651.
[3] Benson D. J., Carlson J. F., and Rickard J., Thick subcategories of the stable module category, Fund. Math. 153 (1997), 5980. MR 1450996.
[4] Benson D. J., Iyengar S. B., and Krause H., Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), 16431684. MR 2846489. DOI 10.4007/ annals.2011.174.3.6.
[5] Bondal A. and Bergh M. van den, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 136, 258. MR 1996800.
[6] Buchweitz R.-O., Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings, preprint, 1987.
[7] Hartshorne R., Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. MR 0463157.
[8] Krause H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), 11281162. MR 2157133. DOI 10.1112/S0010437X05001375.
[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.
[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), 547566. MR 1191736.
[11] Neeman A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205236. MR 1308405. DOI 10.1090/ S0894-0347-96-00174-9. ˇ
[12] Oppermann S. and Št’ov´íček J., Generating the bounded derived category and perfect ghosts, Bull. Lond. Math. Soc. 44 (2012), 285298. MR 2914607. DOI 10.1112/blms/ bdr093.
[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., 240262; English translation in Proc. Steklov Inst. Math. 246 (2004), 227248. MR 2101296.
[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, 85147. MR 0338129.
[15] Schoutens H., Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217239. MR 1969220. DOI 10.1081/AGB-120016756.
[16] Stevenson G., Subcategories of singularity categories via tensor actions, preprint, arXiv:1105.4698v3 [math.AG]
[17] Takahashi R., Classifying thick subcategories of the stable category of Cohen- Macaulay modules, Adv. Math. 225 (2010), 20762116. MR 2680200. DOI 10.1016/ j.aim.2010.04.009.
[18] Takahashi R., Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the punctured spectra, J. Math. Soc. Japan 65 (2013), 357374.
[19] Verdier J.-L., Des catégories dérivées des catégories abéliennes, Astérisque 239, Soc. Math. France, Paris, 1997. MR 1453167.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 14 *
Loading metrics...

Abstract views

Total abstract views: 82 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th January 2018. This data will be updated every 24 hours.