# 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

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topBenson, 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] 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] L. Alonso Tarrío, A. Jeremías López & M. J. Souto Salorio, Bousfield localization on formal schemes, J. Algebra278 (2004), 585–610. Zbl1060.18007
- [3] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math.96 (1989), 71–101. Zbl0677.13004MR981738
- [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] 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] L. L. Avramov & R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math.142 (2000), 285–318. Zbl0999.13008
- [7] L. L. Avramov, H.-B. Foxby & S. Halperin, Differential graded homological algebra, in preparation.
- [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] L. L. Avramov & S. B. Iyengar, Constructing modules with prescribed cohomological support, Illinois J. Math.51 (2007), 1–20. Zbl1121.13014
- [10] L. L. Avramov & L.-C. Sun, Cohomology operators defined by a deformation, J. Algebra204 (1998), 684–710. Zbl0915.13009
- [11] P. Balmer, Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math.129 (2007), 1227–1250. Zbl1130.18005MR2354319
- [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] D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, 1991. Zbl0731.20001MR1156302
- [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] D. J. Benson & J. F. Carlson, Varieties and cohomology of infinitely generated modules, Arch. Math. (Basel) 91 (2008), 122–125. Zbl1151.20040
- [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] D. J. Benson, S. B. Iyengar & H. Krause, Stratifying modular representations of finite groups, preprint, 2008. Zbl1261.20057
- [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] D. J. Benson & H. Krause, Complexes of injective $kG$-modules, Algebra Number Theory2 (2008), 1–30. Zbl1167.20006
- [20] P. A. Bergh, On support varieties for modules over complete intersections, Proc. Amer. Math. Soc.135 (2007), 3795–3803. Zbl1127.13008MR2341929
- [21] W. Bruns & J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993. Zbl0788.13005
- [22] J. F. Carlson, The varieties and the cohomology ring of a module, J. Algebra85 (1983), 104–143. Zbl0526.20040MR723070
- [23] J. F. Carlson, The variety of an indecomposable module is connected, Invent. Math.77 (1984), 291–299. Zbl0543.20032MR752822
- [24] H. Cartan & S. Eilenberg, Homological algebra, Princeton University Press, 1956. Zbl0933.18001
- [25] S. K. Chebolu, Krull-Schmidt decompositions for thick subcategories, J. Pure Appl. Algebra210 (2007), 11–27. Zbl1116.55004MR2311169
- [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] H.-B. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra15 (1979), 149–172. Zbl0411.13006MR535182
- [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] E. M. Friedlander & J. Pevtsova, $\Pi $-supports for modules for finite group schemes, Duke Math. J.139 (2007), 317–368. Zbl1128.20031
- [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] J. P. C. Greenlees & J. P. May, Derived functors of $I$-adic completion and local homology, J. Algebra149 (1992), 438–453. Zbl0774.18007
- [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] 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] 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] 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] M. Hovey, J. H. Palmieri & N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), 114. Zbl0881.55001
- [37] S. B. Iyengar & H. Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math.11 (2006), 207–240. Zbl1119.13014
- [38] H. Krause, Decomposing thick subcategories of the stable module category, Math. Ann.313 (1999), 95–108. Zbl0926.20004MR1666825
- [39] H. Krause, A Brown representability theorem via coherent functors, Topology41 (2002), 853–861. Zbl1009.18010MR1905842
- [40] H. Krause, The stable derived category of a Noetherian scheme, Compos. Math.141 (2005), 1128–1162. Zbl1090.18006MR2157133
- [41] H. Krause, Thick subcategories of modules over commutative Noetherian rings, Math. Ann.340 (2008), 733–747. Zbl1143.13012MR2372735
- [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] 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] S. MacLane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer, 1971. Zbl0232.18001MR354798
- [45] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library 29, North-Holland Publishing Co., 1983. Zbl0552.55002MR738973
- [46] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986. Zbl0603.13001MR879273
- [47] A. Neeman, The chromatic tower for $D\left(R\right)$, Topology31 (1992), 519–532. Zbl0793.18008MR1174255
- [48] A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, 2001. Zbl0974.18008MR1812507
- [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] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc.56 (1997), 149–170. Zbl0910.20034MR1462832
- [51] U. Shukla, Cohomologie des algèbres associatives, Ann. Sci. École Norm. Sup.78 (1961), 163–209. Zbl0228.18005MR132769
- [52] N. Snashall & Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004), 705–732. Zbl1067.16010
- [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] R. W. Thomason, The classification of triangulated subcategories, Compositio Math.105 (1997), 1–27. Zbl0873.18003MR1436741

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