Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Tensor ideals and varieties for modules of quantum elementary abelian groups

Authors: Julia Pevtsova and Sarah Witherspoon
Journal: Proc. Amer. Math. Soc. 143 (2015), 3727-3741
MSC (2010): Primary 16E40, 16T05, 18D10
Published electronically: April 6, 2015
MathSciNet review: 3359565
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a previous paper we constructed rank and support variety theories for ``quantum elementary abelian groups,'' that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.

References [Enhancements On Off] (What's this?)

  • [1] George S. Avrunin and Leonard L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), no. 2, 277-286. MR 656624 (83h:20048),
  • [2] D. J. Benson, Representations and cohomology, Volumes I and II, Cambridge University Press, 1991.
  • [3] 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 (97f:20008),
  • [4] 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 (98g:20021)
  • [5] David J. Benson, Karin Erdmann, and Miles Holloway, Rank varieties for a class of finite-dimensional local algebras, J. Pure Appl. Algebra 211 (2007), no. 2, 497-510. MR 2341266 (2008f:16006),
  • [6] D. J. Benson and E. L. Green, Non-principal blocks with one simple module, Q. J. Math. 55 (2004), no. 1, 1-11. MR 2043003 (2004k:20017),
  • [7] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Localising subcategories for cochains on the classifying space of a finite group, C. R. Math. Acad. Sci. Paris 349 (2011), no. 17-18, 953-956 (English, with English and French summaries). MR 2838242 (2012i:20067),
  • [8] D. J. Benson and S. Witherspoon, Examples of support varieties for Hopf algebras with noncommutative tensor products, arxiv:1308.5262.
  • [9] Aslak Bakke Buan, Henning Krause, and Øyvind Solberg, Support varieties: an ideal approach, Homology, Homotopy Appl. 9 (2007), no. 1, 45-74. MR 2280286 (2008i:18007)
  • [10] Jon F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), no. 2, 291-299. MR 752822 (86b:20009),
  • [11] J. F. Carlson and S. Iyengar, Thick subcategories of the bounded derived category of a finite group, arxiv:1201.6536.
  • [12] D. Fischman, S. Montgomery, and H.-J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4857-4895. MR 1401518 (98c:16049),
  • [13] 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 (2008g:14081),
  • [14] Victor Ginzburg and Shrawan Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179-198. MR 1201697 (94c:17026),
  • [15] 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 (89e:16035)
  • [16] 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 (89g:55022)
  • [17] 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 (98a:55017),
  • [18] Richard Gustavus Larson and Moss Eisenberg Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75-94. MR 0240169 (39 #1523)
  • [19] M. Mastnak, J. Pevtsova, P. Schauenburg, and S. Witherspoon, Cohomology of finite-dimensional pointed Hopf algebras, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 377-404. MR 2595743 (2011b:16116),
  • [20] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507 (2001k:18010)
  • [21] Amnon Neeman, The chromatic tower for $ D(R)$, Topology 31 (1992), no. 3, 519-532. With an appendix by Marcel Bökstedt. MR 1174255 (93h:18018),
  • [22] Julia Pevtsova and Sarah Witherspoon, Varieties for modules of quantum elementary abelian groups, Algebr. Represent. Theory 12 (2009), no. 6, 567-595. MR 2563183 (2011b:16035),
  • [23] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16E40, 16T05, 18D10

Retrieve articles in all journals with MSC (2010): 16E40, 16T05, 18D10

Additional Information

Julia Pevtsova
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Sarah Witherspoon
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Support variety, rank variety, elementary abelian group, tensor product property, tensor ideal
Received by editor(s): December 18, 2013
Received by editor(s) in revised form: April 11, 2014
Published electronically: April 6, 2015
Additional Notes: This material is based upon work supported by the National Science Foundation under grant No. 0932078000, while the second author was in residence at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, during the semester of Spring 2013. The first author was supported by NSF grant DMS-0953011, and the second author by NSF grant DMS-1101399.
Communicated by: Kailash Misra
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society