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Stratification for module categories of finite group schemes


Authors: Dave Benson, Srikanth B. Iyengar, Henning Krause and Julia Pevtsova
Journal: J. Amer. Math. Soc. 31 (2018), 265-302
MSC (2010): Primary 16G10; Secondary 20G10, 20J06
DOI: https://doi.org/10.1090/jams/887
Published electronically: August 15, 2017
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Abstract: The tensor ideal localizing subcategories of the stable module category of all, including infinite dimensional, representations of a finite group scheme over a field of positive characteristic are classified. Various applications concerning the structure of the stable module category and the behavior of support and cosupport under restriction and induction are presented.


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  • [1] Luchezar L. Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285-318. MR 1794064, https://doi.org/10.1007/s002220000090
  • [2] Christopher P. Bendel, Cohomology and projectivity of modules for finite group schemes, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 3, 405-425. MR 1866385
  • [3] D. J. Benson, Representations and cohomology. II, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1998. Cohomology of groups and modules. MR 1634407
  • [4] D. J. Benson, Jon F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 4, 597-615. MR 1401950, https://doi.org/10.1017/S0305004100001584
  • [5] D. J. Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59-80. MR 1450996
  • [6] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 575-621 (English, with English and French summaries). MR 2489634, https://doi.org/10.24033/asens.2076
  • [7] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying triangulated categories, J. Topol. 4 (2011), no. 3, 641-666. MR 2832572, https://doi.org/10.1112/jtopol/jtr017
  • [8] David J. Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), no. 3, 1643-1684. MR 2846489, https://doi.org/10.4007/annals.2011.174.3.6
  • [9] David J. Benson, Srikanth B. Iyengar, and Henning Krause, Colocalizing subcategories and cosupport, J. Reine Angew. Math. 673 (2012), 161-207. MR 2999131
  • [10] Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova, Colocalising subcategories of modules over finite group schemes, Ann. K-Theory 2 (2017), no. 3, 387-408. MR 3658989, https://doi.org/10.2140/akt.2017.2.387
  • [11] D. J. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, Stratification and $ \pi $-cosupport: Finite groups, Math. Z., to appear.
  • [12] David J. Benson, Srikanth Iyengar, and Henning Krause, Representations of finite groups: local cohomology and support, Oberwolfach Seminars, vol. 43, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2951763
  • [13] David Benson and Henning Krause, Pure injectives and the spectrum of the cohomology ring of a finite group, J. Reine Angew. Math. 542 (2002), 23-51. MR 1880824, https://doi.org/10.1515/crll.2002.008
  • [14] David John Benson and Henning Krause, Complexes of injective $ kG$-modules, Algebra Number Theory 2 (2008), no. 1, 1-30. MR 2377361, https://doi.org/10.2140/ant.2008.2.1
  • [15] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257-281. MR 551009, https://doi.org/10.1016/0040-9383(79)90018-1
  • [16] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1998. MR 1251956
  • [17] Jesse Burke, Finite injective dimension over rings with Noetherian cohomology, Math. Res. Lett. 19 (2012), no. 4, 741-752. MR 3008411, https://doi.org/10.4310/MRL.2012.v19.n4.a1
  • [18] Jon F. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), no. 1, 104-143. MR 723070, https://doi.org/10.1016/0021-8693(83)90121-7
  • [19] Jon F. Carlson, Modules and group algebras, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR 1393196
  • [20] Leo G. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), no. 3, 287-302. MR 0401943, https://doi.org/10.1016/0022-4049(76)90055-4
  • [21] Everett Dade, Endo-permutation modules over $ p$-groups. II, Ann. of Math. (2) 108 (1978), no. 2, 317-346. MR 506990, https://doi.org/10.2307/1971169
  • [22] Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), no. 2, 207-241. MR 960945, https://doi.org/10.2307/1971440
  • [23] Eric M. Friedlander and Julia Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005), no. 2, 379-420. MR 2130619
  • [24] Eric M. Friedlander and Julia Pevtsova, $ \Pi$-supports for modules for finite group schemes, Duke Math. J. 139 (2007), no. 2, 317-368. MR 2352134, https://doi.org/10.1215/S0012-7094-07-13923-1
  • [25] Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209-270. MR 1427618, https://doi.org/10.1007/s002220050119
  • [26] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124
  • [27] Michael J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73-96. MR 932260
  • [28] Mark Hovey and John H. Palmieri, Stably thick subcategories of modules over Hopf algebras, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 3, 441-474. MR 1816804, https://doi.org/10.1017/S0305004101005060
  • [29] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895, https://doi.org/10.1090/memo/0610
  • [30] Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
  • [31] Amnon Neeman, The chromatic tower for $ D(R)$, Topology 31 (1992), no. 3, 519-532. With an appendix by Marcel Bökstedt. MR 1174255, https://doi.org/10.1016/0040-9383(92)90047-L
  • [32] Amnon Neeman, 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), no. 5, 547-566. MR 1191736
  • [33] Amnon Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205-236. MR 1308405, https://doi.org/10.1090/S0894-0347-96-00174-9
  • [34] Julia Pevtsova, Infinite dimensional modules for Frobenius kernels, J. Pure Appl. Algebra 173 (2002), no. 1, 59-86. MR 1912960, https://doi.org/10.1016/S0022-4049(01)00168-2
  • [35] Julia Pevtsova, Support cones for infinitesimal group schemes, Hopf algebras, Lecture Notes in Pure and Appl. Math., vol. 237, Dekker, New York, 2004, pp. 203-213. MR 2051741
  • [36] Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351-414. MR 737778, https://doi.org/10.2307/2374308
  • [37] Andrei Suslin, Detection theorem for finite group schemes, J. Pure Appl. Algebra 206 (2006), no. 1-2, 189-221. MR 2220088, https://doi.org/10.1016/j.jpaa.2005.03.017
  • [38] Andrei Suslin, Eric M. Friedlander, and Christopher P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729-759. MR 1443547, https://doi.org/10.1090/S0894-0347-97-00239-7
  • [39] R. W. Thomason, The classification of triangulated subcategories, Compo. Math. 105 (1997), no. 1, 1-27. MR 1436741, https://doi.org/10.1023/A:1017932514274
  • [40] William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117

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Additional Information

Dave Benson
Affiliation: Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland, United Kingdom

Srikanth B. Iyengar
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Henning Krause
Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Julia Pevtsova
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: julia@math.washington.edu

DOI: https://doi.org/10.1090/jams/887
Keywords: Cosupport, finite group scheme, localizing subcategory, support, thick subcategory
Received by editor(s): November 3, 2015
Received by editor(s) in revised form: April 5, 2016, April 24, 2017, and June 16, 2017
Published electronically: August 15, 2017
Additional Notes: The second author was partly supported by NSF grant DMS 1503044.
The fourth author was partly supported by NSF grant DMS 0953011.
Article copyright: © Copyright 2017 American Mathematical Society

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