Local cohomology and support for triangulated categories

Dave Benson; Srikanth B. Iyengar; Henning Krause

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 4, page 575-621
  • ISSN: 0012-9593

Abstract

top
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.

How to cite

top

Benson, Dave, Iyengar, Srikanth B., and Krause, Henning. "Local cohomology and support for triangulated categories." Annales scientifiques de l'École Normale Supérieure 41.4 (2008): 575-621. <http://eudml.org/doc/272216>.

@article{Benson2008,
abstract = {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.},
author = {Benson, Dave, Iyengar, Srikanth B., Krause, Henning},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {local cohomology; support; triangulated category; complete intersection},
language = {eng},
number = {4},
pages = {575-621},
publisher = {Société mathématique de France},
title = {Local cohomology and support for triangulated categories},
url = {http://eudml.org/doc/272216},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Benson, Dave
AU - Iyengar, Srikanth B.
AU - Krause, Henning
TI - Local cohomology and support for triangulated categories
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 4
SP - 575
EP - 621
AB - 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.
LA - eng
KW - local cohomology; support; triangulated category; complete intersection
UR - http://eudml.org/doc/272216
ER -

References

top
  1. [1] L. Alonso Tarrío, A. Jeremías López & M. J. Souto Salorio, Localization in categories of complexes and unbounded resolutions, Canad. J. Math.52 (2000), 225–247. Zbl0948.18008
  2. [2] L. Alonso Tarrío, A. Jeremías López & M. J. Souto Salorio, Bousfield localization on formal schemes, J. Algebra278 (2004), 585–610. Zbl1060.18007
  3. [3] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math.96 (1989), 71–101. Zbl0677.13004MR981738
  4. [4] L. L. Avramov, Infinite free resolutions, in Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math. 166, Birkhäuser, 1998, 1–118. Zbl0934.13008MR1648664
  5. [5] L. L. Avramov & R.-O. Buchweitz, Homological algebra modulo a regular sequence with special attention to codimension two, J. Algebra230 (2000), 24–67. Zbl1011.13007
  6. [6] L. L. Avramov & R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math.142 (2000), 285–318. Zbl0999.13008
  7. [7] L. L. Avramov, H.-B. Foxby & S. Halperin, Differential graded homological algebra, in preparation. 
  8. [8] L. L. Avramov, V. N. Gasharov & I. V. Peeva, Complete intersection dimension, Publ. Math. I.H.É.S. 86 (1997), 67–114. Zbl0918.13008
  9. [9] L. L. Avramov & S. B. Iyengar, Constructing modules with prescribed cohomological support, Illinois J. Math.51 (2007), 1–20. Zbl1121.13014
  10. [10] L. L. Avramov & L.-C. Sun, Cohomology operators defined by a deformation, J. Algebra204 (1998), 684–710. Zbl0915.13009
  11. [11] P. Balmer, Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math.129 (2007), 1227–1250. Zbl1130.18005MR2354319
  12. [12] A. A. Beĭlinson, J. Bernstein & P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, 1982, 5–171. Zbl0536.14011
  13. [13] D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, 1991. Zbl0731.20001MR1156302
  14. [14] D. J. Benson, Commutative algebra in the cohomology of groups, in Trends in commutative algebra, Math. Sci. Res. Inst. Publ. 51, Cambridge Univ. Press, 2004, 1–50. Zbl1113.20042MR2132647
  15. [15] D. J. Benson & J. F. Carlson, Varieties and cohomology of infinitely generated modules, Arch. Math. (Basel) 91 (2008), 122–125. Zbl1151.20040
  16. [16] D. J. Benson, J. F. Carlson & J. Rickard, Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120 (1996), 597–615. Zbl0888.20003
  17. [17] D. J. Benson, S. B. Iyengar & H. Krause, Stratifying modular representations of finite groups, preprint, 2008. Zbl1261.20057
  18. [18] D. J. Benson & H. Krause, Pure injectives and the spectrum of the cohomology ring of a finite group, J. reine angew. Math. 542 (2002), 23–51. Zbl0987.20026
  19. [19] D. J. Benson & H. Krause, Complexes of injective k G -modules, Algebra Number Theory2 (2008), 1–30. Zbl1167.20006
  20. [20] P. A. Bergh, On support varieties for modules over complete intersections, Proc. Amer. Math. Soc.135 (2007), 3795–3803. Zbl1127.13008MR2341929
  21. [21] W. Bruns & J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993. Zbl0788.13005
  22. [22] J. F. Carlson, The varieties and the cohomology ring of a module, J. Algebra85 (1983), 104–143. Zbl0526.20040MR723070
  23. [23] J. F. Carlson, The variety of an indecomposable module is connected, Invent. Math.77 (1984), 291–299. Zbl0543.20032MR752822
  24. [24] H. Cartan & S. Eilenberg, Homological algebra, Princeton University Press, 1956. Zbl0933.18001
  25. [25] S. K. Chebolu, Krull-Schmidt decompositions for thick subcategories, J. Pure Appl. Algebra210 (2007), 11–27. Zbl1116.55004MR2311169
  26. [26] K. Erdmann, M. Holloway, R. Taillefer, N. Snashall & Ø. Solberg, Support varieties for selfinjective algebras, K -Theory 33 (2004), 67–87. Zbl1116.16007
  27. [27] H.-B. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra15 (1979), 149–172. Zbl0411.13006MR535182
  28. [28] H.-B. Foxby & S. B. Iyengar, Depth and amplitude for unbounded complexes, in Commutative algebra (Grenoble / Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., 2003, 119–137. Zbl1096.13516
  29. [29] E. M. Friedlander & J. Pevtsova, Π -supports for modules for finite group schemes, Duke Math. J.139 (2007), 317–368. Zbl1128.20031
  30. [30] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer New York, Inc., New York, 1967. Zbl0186.56802
  31. [31] J. P. C. Greenlees & J. P. May, Derived functors of I -adic completion and local homology, J. Algebra149 (1992), 438–453. Zbl0774.18007
  32. [32] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents mathématiques, 4, Soc. Math. France, 2005, Séminaire de Géométrie Algébrique du Bois Marie, 1962. Zbl1079.14001MR2171939
  33. [33] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand.34 (1974), 167–183. Zbl0292.13009MR364232
  34. [34] R. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. Lecture Notes in Math., No. 20, Springer, 1966. MR222093
  35. [35] M. J. Hopkins, Global methods in homotopy theory, in Homotopy theory (Durham, 1985), London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, 1987, 73–96. Zbl0657.55008MR932260
  36. [36] M. Hovey, J. H. Palmieri & N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), 114. Zbl0881.55001
  37. [37] S. B. Iyengar & H. Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math.11 (2006), 207–240. Zbl1119.13014
  38. [38] H. Krause, Decomposing thick subcategories of the stable module category, Math. Ann.313 (1999), 95–108. Zbl0926.20004MR1666825
  39. [39] H. Krause, A Brown representability theorem via coherent functors, Topology41 (2002), 853–861. Zbl1009.18010MR1905842
  40. [40] H. Krause, The stable derived category of a Noetherian scheme, Compos. Math.141 (2005), 1128–1162. Zbl1090.18006MR2157133
  41. [41] H. Krause, Thick subcategories of modules over commutative Noetherian rings, Math. Ann.340 (2008), 733–747. Zbl1143.13012MR2372735
  42. [42] L. G. J. Lewis, J. P. May, M. Steinberger & J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, 1986. Zbl0611.55001
  43. [43] J. Lipman, Lectures on local cohomology and duality, in Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Appl. Math. 226, Dekker, 2002, 39–89. Zbl1011.13010MR1888195
  44. [44] S. MacLane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer, 1971. Zbl0232.18001MR354798
  45. [45] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library 29, North-Holland Publishing Co., 1983. Zbl0552.55002MR738973
  46. [46] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986. Zbl0603.13001MR879273
  47. [47] A. Neeman, The chromatic tower for D ( R ) , Topology31 (1992), 519–532. Zbl0793.18008MR1174255
  48. [48] A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, 2001. Zbl0974.18008MR1812507
  49. [49] D. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. 94 (1971), 549–572; ibid. 94 (1971), 573–602. Zbl0247.57013MR298694
  50. [50] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc.56 (1997), 149–170. Zbl0910.20034MR1462832
  51. [51] U. Shukla, Cohomologie des algèbres associatives, Ann. Sci. École Norm. Sup.78 (1961), 163–209. Zbl0228.18005MR132769
  52. [52] N. Snashall & Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004), 705–732. Zbl1067.16010
  53. [53] Ø. Solberg, Support varieties for modules and complexes, in Trends in representation theory of algebras and related topics, Contemp. Math. 406, Amer. Math. Soc., 2006, 239–270. Zbl1115.16007MR2258047
  54. [54] R. W. Thomason, The classification of triangulated subcategories, Compositio Math.105 (1997), 1–27. Zbl0873.18003MR1436741

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.