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The ring of polynomial functors of prime degree


Author: Alexander Zimmermann
Journal: Trans. Amer. Math. Soc. 367 (2015), 7161-7192
MSC (2010): Primary 16H10; Secondary 20C30, 20J06, 55R40
DOI: https://doi.org/10.1090/S0002-9947-2014-06265-X
Published electronically: December 17, 2014
MathSciNet review: 3378827
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Abstract: Let $ \hat {\mathbb{Z}}_p$ be the ring of $ p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $ \hat {\mathbb{Z}}_p$-modules of degree at most $ p$ is equivalent to the category of modules over a particularly well understood ring, called Green order. This case was conjectured by Yuri Drozd.


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  • [1] Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
  • [2] Hans Joachim Baues, Quadratic functors and metastable homotopy, J. Pure Appl. Algebra 91 (1994), no. 1-3, 49-107. MR 1255923 (94j:55022), https://doi.org/10.1016/0022-4049(94)90135-X
  • [3] Hans-Joachim Baues, Winfried Dreckmann, Vincent Franjou, and Teimuraz Pirashvili, Foncteurs polynomiaux et foncteurs de Mackey non linéaires, Bull. Soc. Math. France 129 (2001), no. 2, 237-257 (French, with English and French summaries). MR 1871297 (2002j:18004)
  • [4] D. J. Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. MR 1110581 (92m:20005)
  • [5] Aurélien Djament, Sur l'homologie des groupes unitaires à coefficients polynomiaux, J. K-Theory 10 (2012), no. 1, 87-139 (French, with English and French summaries). MR 2990563, https://doi.org/10.1017/is012003003jkt184
  • [6] Aurélien Djament and Christine Vespa, Sur l'homologie des groupes orthogonaux et symplectiques à coefficients tordus, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 3, 395-459 (French, with English and French summaries). MR 2667021 (2011h:20095)
  • [7] Aurélien Djament and Christine Vespa, Sur l'homologie des groupes d'automorphismes des groupes libres à coefficients polynomiaux, preprint 2013, arXiv: 1210.4030v2
  • [8] Yuriy A. Drozd, Finitely generated quadratic modules, Manuscripta Math. 104 (2001), no. 2, 239-256. MR 1821185 (2002d:16021), https://doi.org/10.1007/s002290170041
  • [9] Yuriy A. Drozd, On cubic functors, Comm. Algebra 31 (2003), no. 3, 1147-1173. MR 1971055 (2004a:16031), https://doi.org/10.1081/AGB-120017759
  • [10] Samuel Eilenberg and Saunders MacLane, On the groups $ H(\Pi ,n)$. III, Ann. of Math. (2) 60 (1954), 513-557. MR 0065163 (16,392a)
  • [11] Vincent Franjou, Extensions entre puissances extérieures et entre puissances symétriques, J. Algebra 179 (1996), no. 2, 501-522 (French). MR 1367860 (96i:20061), https://doi.org/10.1006/jabr.1996.0022
  • [12] Vincent Franjou, Jean Lannes, and Lionel Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994), no. 3, 513-538 (French, with English and French summaries). MR 1262942 (95d:19002), https://doi.org/10.1007/BF01231771
  • [13] Vincent Franjou and Teimuraz Pirashvili, On the Mac Lane cohomology for the ring of integers, Topology 37 (1998), no. 1, 109-114. MR 1480880 (98h:19002), https://doi.org/10.1016/S0040-9383(97)00005-0
  • [14] Vincent Franjou and Teimuraz Pirashvili, Stable $ K$-theory is bifunctor homology (after A. Scorichenko), Rational representations, the Steenrod algebra and functor homology, Panor. Synthèses, vol. 16, Soc. Math. France, Paris, 2003, pp. 107-126 (English, with English and French summaries). MR 2117530
  • [15] Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin, General linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (1999), no. 2, 663-728. MR 1726705 (2001b:14076), https://doi.org/10.2307/121092
  • [16] Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209-270. MR 1427618 (98h:14055a), https://doi.org/10.1007/s002220050119
  • [17] J. A. Green, Polynomial representations of $ {\rm GL}_{n}$, Second corrected and augmented edition, Lecture Notes in Mathematics, vol. 830, Springer, Berlin, 2007. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. Schocker. MR 2349209 (2009b:20084)
  • [18] Manfred Hartl, Teimuraz Pirashvili and Christine Vespa, Polynomial functors from algebras over a set-operad and non-linear Mackey functors, preprint (2012) arxiv:1209.1607v2
  • [19] Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. J. Math. 115 (1993), no. 5, 1053-1106. MR 1246184 (94i:55024), https://doi.org/10.2307/2375065
  • [20] Steffen König, Cyclotomic Schur algebras and blocks of cyclic defect, Canad. Math. Bull. 43 (2000), no. 1, 79-86. MR 1749952 (2001f:20022), https://doi.org/10.4153/CMB-2000-012-0
  • [21] Steffen König and Alexander Zimmermann, Derived equivalences for group rings, Lecture Notes in Mathematics, vol. 1685, Springer-Verlag, Berlin, 1998. With contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier. MR 1649837 (2000g:16018)
  • [22] Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (1994), no. 2, 327-360. MR 1269607 (95c:55022), https://doi.org/10.2307/2374932
  • [23] Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II, $ K$-Theory 8 (1994), no. 4, 395-428. MR 1300547 (95k:55038), https://doi.org/10.1007/BF00961409
  • [24] Nicholas J. Kuhn, The generic representation theory of finite fields: a survey of basic structure, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 193-212. MR 1789216 (2001m:20065)
  • [25] Nicholas J. Kuhn, A stratification of generic representation theory and generalized Schur algebras, $ K$-Theory 26 (2002), no. 1, 15-49. MR 1918209 (2003h:20089), https://doi.org/10.1023/A:1016357323204
  • [26] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872 (2001j:18001)
  • [27] T. I. Pirashvili, Polynomial functors, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 55-66 (Russian, with English summary). MR 1029007 (91f:18004)
  • [28] Teimuraz Pirashvili, Polynomial functors over finite fields (after Franjou, Friedlander, Henn, Lannes, Schwartz, Suslin), Astérisque 276 (2002), 369-388. Séminaire Bourbaki, Vol. 1999/2000. MR 1886766 (2003c:20053)
  • [29] Teimuraz Pirashvili, Introduction to functor homology, Rational representations, the Steenrod algebra and functor homology, Panor. Synthèses, vol. 16, Soc. Math. France, Paris, 2003, pp. 1-26 (English, with English and French summaries). MR 2117526
  • [30] Laurent Piriou, Extensions entre foncteurs de la catégorie des espaces vectoriels sur le corps premier à $ p$ éléments dans elle-même, thèse de doctorat université Paris 7 (1995).
  • [31] Laurent Piriou and Lionel Schwartz, Extensions de foncteurs simples, $ K$-Theory 15 (1998), no. 3, 269-291 (French, with English summary). MR 1659961 (2000g:20085), https://doi.org/10.1023/A:1007700132156
  • [32] Chrysostomos Psaroudakis and Jorge Vitória, Recollements of module categories, Appl. Categ. Structures 22 (2014), no. 4, 579-593. MR 3227608, https://doi.org/10.1007/s10485-013-9323-x
  • [33] I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original. With a foreword by M. J. Taylor. MR 1972204 (2004c:16026)
  • [34] K. W. Roggenkamp, Blocks of cyclic defect and Green-orders, Comm. Algebra 20 (1992), no. 6, 1715-1734. MR 1162603 (93g:20014), https://doi.org/10.1080/00927879208824426
  • [35] Antoine Touzé, Cohomologie Rationnelle du Groupe Linéaire et Extensions de Bifoncteurs, Thèse de doctorat de l'université de Nantes (2008).
  • [36] Antoine Touzé and Wilberd van der Kallen, Bifunctor cohomology and cohomological finite generation for reductive groups, Duke Math. J. 151 (2010), no. 2, 251-278. MR 2598378 (2011g:20075), https://doi.org/10.1215/00127094-2009-065
  • [37] Christine Vespa, Generic representations of orthogonal groups: the mixed functors, Algebr. Geom. Topol. 7 (2007), 379-410. MR 2308951 (2008c:18002), https://doi.org/10.2140/agt.2007.7.379
  • [38] Christine Vespa, Generic representations of orthogonal groups: the functor category $ {\mathcal {F}}_{\rm quad}$, J. Pure Appl. Algebra 212 (2008), no. 6, 1472-1499. MR 2391661 (2008m:18001), https://doi.org/10.1016/j.jpaa.2007.10.014
  • [39] Christine Vespa, Generic representations of orthogonal groups: projective functors in the category $ \mathcal {F}_{\rm quad}$, Fund. Math. 200 (2008), no. 3, 243-278. MR 2429598 (2009i:18001), https://doi.org/10.4064/fm200-3-2
  • [40] Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727 (95d:55017)

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Additional Information

Alexander Zimmermann
Affiliation: Département de Mathématiques et CNRS UMR 7352, Université de Picardie, 33 rue St Leu, F-80039 Amiens Cedex 1, France
Email: alexander.zimmermann@u-picardie.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06265-X
Keywords: Polynomial functors, Green orders, Brauer tree algebras, Schur algebras, recollement diagram, representation type
Received by editor(s): April 17, 2013
Received by editor(s) in revised form: July 29, 2013, and August 12, 2013
Published electronically: December 17, 2014
Additional Notes: This research was supported by a grant “PAI alliance” from the Ministère des Affaires Étrangères de France and the British Council. The author acknowledges support from STIC Asie of the Ministère des Affaires Étrangères de France
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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