Local cohomology and support for triangulated categories
[Cohomologie locale et support pour les catégories triangulées]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 575-621.

Nous proposons une façon nouvelle de définir une notion de support pour les objets d'une catégorie avec petits coproduits, engendrée par des objets compacts. Cette approche est basée sur une construction des foncteurs de cohomologie locale sur les catégories triangulées relativement à un anneau central d'opérateurs. Comme cas particuliers, on retrouve la théorie pour les anneaux noethériens de Foxby et Neeman, la théorie d'Avramov et Buchweitz pour les anneaux locaux d'intersection complète, ou les variétés pour les représentations des groupes finis selon Benson, Carlson et Rickard. Nous donnons des exemples explicites d'objets dont le support triangulé et le support cohomologique diffèrent. Dans le cas des représentations des groupes, ceci nous permet de corriger et d'établir une conjecture de Benson.

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Special cases are, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects, the triangulated support and cohomological support of which differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson.

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Benson, Dave; Iyengar, Srikanth B.; Krause, Henning. Local cohomology and support for triangulated categories. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 575-621. doi : 10.24033/asens.2076. http://archive.numdam.org/articles/10.24033/asens.2076/

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  • Ben-Zvi, David; Nadler, David; Preygel, Anatoly A spectral incarnation of affine character sheaves, Compositio Mathematica, Volume 153 (2017) no. 9, pp. 1908-1944 | DOI:10.1112/s0010437x17007278 | Zbl:1439.20004
  • Sather-Wagstaff, Sean; Wicklein, Richard Adically finite chain complexes, Journal of Algebra and its Applications, Volume 16 (2017) no. 12, p. 23 (Id/No 1750232) | DOI:10.1142/s0219498817502322 | Zbl:1394.13010
  • Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia Stratification and π-cosupport: finite groups, Mathematische Zeitschrift, Volume 287 (2017) no. 3-4, pp. 947-965 | DOI:10.1007/s00209-017-1853-8 | Zbl:1483.20088
  • Sather-Wagstaff, Sean; Wicklein, Richard Extended local cohomology and local homology, Algebras and Representation Theory, Volume 19 (2016) no. 5, pp. 1217-1238 | DOI:10.1007/s10468-016-9616-5 | Zbl:1360.13025
  • Liu, Yanping; Liu, Zhongkui; Yang, Xiaoyan Depth for triangulated categories, Bulletin of the Korean Mathematical Society, Volume 53 (2016) no. 2, pp. 551-559 | DOI:10.4134/bkms.2016.53.2.551 | Zbl:1346.18017
  • Banerjee, Abhishek Realizations of pairs and Oka families in tensor triangulated categories, European Journal of Mathematics, Volume 2 (2016) no. 3, pp. 760-797 | DOI:10.1007/s40879-016-0108-2 | Zbl:1361.13002
  • Mathew, Akhil A thick subcategory theorem for modules over certain ring spectra, Geometry Topology, Volume 19 (2015) no. 4, pp. 2359-2392 | DOI:10.2140/gt.2015.19.2359 | Zbl:1405.55009
  • Benson, Dave; Iyengar, Srikanth B.; Krause, Henning A local-global principle for small triangulated categories, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 158 (2015) no. 3, pp. 451-476 | DOI:10.1017/s0305004115000067 | Zbl:1371.18008
  • Arinkin, D.; Gaitsgory, D. Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Mathematica. New Series, Volume 21 (2015) no. 1, pp. 1-199 | DOI:10.1007/s00029-014-0167-5 | Zbl:1423.14085
  • Burke, Jesse; Walker, Mark E. Matrix factorizations in higher codimension, Transactions of the American Mathematical Society, Volume 367 (2015) no. 5, pp. 3323-3370 | DOI:10.1090/s0002-9947-2015-06323-5 | Zbl:1394.13016
  • Shamir, Shoham On the string topology category of compact Lie groups, Advances in Mathematics, Volume 261 (2014), pp. 122-153 | DOI:10.1016/j.aim.2014.05.002 | Zbl:1309.55006
  • Benson, Dave; Witherspoon, Sarah Examples of support varieties for Hopf algebras with noncommutative tensor products., Archiv der Mathematik, Volume 102 (2014) no. 6, pp. 513-520 | DOI:10.1007/s00013-014-0659-8 | Zbl:1310.16025
  • Xu, Fei Spectra of tensor triangulated categories over category algebras, Archiv der Mathematik, Volume 103 (2014) no. 3, pp. 235-253 | DOI:10.1007/s00013-014-0684-7 | Zbl:1309.18014
  • Bergh, Petter Andreas; Jorgensen, David A. On growth in minimal totally acyclic complexes, Journal of Commutative Algebra, Volume 6 (2014) no. 1, pp. 17-31 | DOI:10.1216/jca-2014-6-1-17 | Zbl:1304.13030
  • Benson, David; Greenlees, John Stratifying the derived category of cochains on BG for G a compact Lie group, Journal of Pure and Applied Algebra, Volume 218 (2014) no. 4, pp. 642-650 | DOI:10.1016/j.jpaa.2013.08.004 | Zbl:1291.18014
  • Dell'Ambrogio, Ivo; Stevenson, Greg On the derived category of a graded commutative noetherian ring, Journal of Algebra, Volume 373 (2013), pp. 356-376 | DOI:10.1016/j.jalgebra.2012.09.038 | Zbl:1272.13002
  • Takahashi, Ryo Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the punctured spectra, Journal of the Mathematical Society of Japan, Volume 65 (2013) no. 2, pp. 357-374 | DOI:10.2969/jmsj/06520357 | Zbl:1314.13029
  • Iyengar, Srikanth B.; Krause, Henning The Bousfield lattice of a triangulated category and stratification, Mathematische Zeitschrift, Volume 273 (2013) no. 3-4, pp. 1215-1241 | DOI:10.1007/s00209-012-1051-7 | Zbl:1328.18017
  • Shamir, Shoham Stratifying derived categories of cochains on certain spaces, Mathematische Zeitschrift, Volume 272 (2012) no. 3-4, pp. 839-868 | DOI:10.1007/s00209-011-0960-1 | Zbl:1271.55010
  • Kuribayashi, Katsuhiko The Hochschild cohomology ring of the singular cochain algebra of a space., Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1779-1805 | DOI:10.5802/aif.2658 | Zbl:1279.16009
  • Benson, David J.; Iyengar, Srikanth B.; Krause, Henning Stratifying modular representations of finite groups., Annals of Mathematics. Second Series, Volume 174 (2011) no. 3, pp. 1643-1684 | DOI:10.4007/annals.2011.174.3.6 | Zbl:1261.20057
  • Benson, Dave; Iyengar, Srikanth B.; Krause, Henning Localising subcategories for cochains on the classifying space of a finite group, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 349 (2011) no. 17-18, pp. 953-956 | DOI:10.1016/j.crma.2011.08.019 | Zbl:1237.18009
  • Asadollahi, Javad; Salarian, Shokrollah; Sazeedeh, Reza On the local cohomology and support for triangulated categories, Kyoto Journal of Mathematics, Volume 51 (2011) no. 4, pp. 811-829 | DOI:10.1215/21562261-1424866 | Zbl:1254.18009
  • Benson, David J.; Iyengar, Srikanth B.; Krause, Henning Module categories for finite group algebras., Representations of algebras and related topics. Proceedings of the 14th international conference on representations of algebras and workshop (ICRA XIV), Tokyo, Japan, August 6–15, 2010., Zürich: European Mathematical Society (EMS), 2011, pp. 55-83 | Zbl:1301.16010
  • Krause, Henning; Šťovíček, Jan The telescope conjecture for hereditary rings via Ext-orthogonal pairs., Advances in Mathematics, Volume 225 (2010) no. 5, pp. 2341-2364 | DOI:10.1016/j.aim.2010.04.027 | Zbl:1242.16007
  • Balmer, Paul Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebraic Geometric Topology, Volume 10 (2010) no. 3, pp. 1521-1563 | DOI:10.2140/agt.2010.10.1521 | Zbl:1204.18005
  • Bergh, Petter Andreas; Iyengar, Srikanth B.; Krause, Henning; Oppermann, Steffen Dimensions of triangulated categories via Koszul objects, Mathematische Zeitschrift, Volume 265 (2010) no. 4, pp. 849-864 | DOI:10.1007/s00209-009-0545-4 | Zbl:1263.18006
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  • Krause, Henning Thick subcategories of modules over commutative noetherian rings (with an appendix by Srikanth Iyengar), Mathematische Annalen, Volume 340 (2008) no. 4, pp. 733-747 | DOI:10.1007/s00208-007-0166-3 | Zbl:1143.13012
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