Tensor ideals and varieties for modules of quantum elementary abelian groups
HTML articles powered by AMS MathViewer
- by Julia Pevtsova and Sarah Witherspoon PDF
- Proc. Amer. Math. Soc. 143 (2015), 3727-3741 Request permission
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
- George S. Avrunin and Leonard L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), no. 2, 277–286. MR 656624, DOI 10.1007/BF01389395
- D. J. Benson, Representations and cohomology, Volumes I and II, Cambridge University Press, 1991.
- 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, DOI 10.1017/S0305004100001584
- 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, DOI 10.4064/fm-153-1-59-80
- 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, DOI 10.1016/j.jpaa.2007.02.006
- 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, DOI 10.1093/qjmath/55.1.1
- 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, DOI 10.1016/j.crma.2011.08.019
- D. J. Benson and S. Witherspoon, Examples of support varieties for Hopf algebras with noncommutative tensor products, arxiv:1308.5262.
- Aslak Bakke Buan, Henning Krause, and Øyvind Solberg, Support varieties: an ideal approach, Homology Homotopy Appl. 9 (2007), no. 1, 45–74. MR 2280286
- Jon F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), no. 2, 291–299. MR 752822, DOI 10.1007/BF01388448
- J. F. Carlson and S. Iyengar, Thick subcategories of the bounded derived category of a finite group, arxiv:1201.6536.
- 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, DOI 10.1090/S0002-9947-97-01814-X
- 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, DOI 10.1215/S0012-7094-07-13923-1
- Victor Ginzburg and Shrawan Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179–198. MR 1201697, DOI 10.1215/S0012-7094-93-06909-8
- 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, DOI 10.1017/CBO9780511629228
- 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
- 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, DOI 10.1090/memo/0610
- Richard Gustavus Larson and Moss Eisenberg Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94. MR 240169, DOI 10.2307/2373270
- 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, DOI 10.1112/plms/pdp030
- Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
- Amnon Neeman, The chromatic tower for $D(R)$, Topology 31 (1992), no. 3, 519–532. With an appendix by Marcel Bökstedt. MR 1174255, DOI 10.1016/0040-9383(92)90047-L
- Julia Pevtsova and Sarah Witherspoon, Varieties for modules of quantum elementary abelian groups, Algebr. Represent. Theory 12 (2009), no. 6, 567–595. MR 2563183, DOI 10.1007/s10468-008-9100-y
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Additional Information
- Julia Pevtsova
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 697536
- Email: julia@math.washington.edu
- Sarah Witherspoon
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 364426
- Email: sjw@math.tamu.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3727-3741
- MSC (2010): Primary 16E40, 16T05, 18D10
- DOI: https://doi.org/10.1090/proc/12524
- MathSciNet review: 3359565