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Koszul, Ringel and Serre duality for strict polynomial functors

Part of: Linear algebraic groups and related topics Abelian categories Homological algebra Categories with structure

Published online by Cambridge University Press:  18 March 2013

Henning Krause
Henning Krause*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany email hkrause@math.uni-bielefeld.de
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Abstract

This is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.

Keywords

strict polynomial functorpolynomial representationSchur algebradivided powerKoszul dualityRingel dualitySerre duality

MSC classification

Secondary: 20G05: Representation theory 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 18E30: Derived categories, triangulated categories 18G10: Resolutions; derived functors 20G10: Cohomology theory 20G43: Schur and $q$-Schur algebras
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 6 , June 2013 , pp. 996 - 1018
Copyright
© The Author(s) 2013 

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References

Akin, K., Extensions of symmetric tensors by alternating tensors, J. Algebra 121 (1989), 358–363.Google Scholar
Akin, K. and Buchsbaum, D. A., Characteristic-free representation theory of the general linear group. II. Homological considerations, Adv. Math. 72 (1988), 171–210.Google Scholar
Akin, K., Buchsbaum, D. A. and Weyman, J., Schur functors and Schur complexes, Adv. Math. 44 (1982), 207–278.Google Scholar
Auslander, M., Representation dimension of Artin algebras, Queen Mary College Mathematics Notes (London, 1971).Google Scholar
Bourbaki, N., Éléments de mathématique. Algèbre. Chapitres 1 à 3 (Hermann, Paris, 1970).Google Scholar
Bourbaki, N., Éléments de mathématique. Algèbre. Chapitres 4 à 7, Lecture Notes in Mathematics, vol. 864 (Masson, Paris, 1981).Google Scholar
Bousfield, A. K., Homogeneous functors and their derived functors, Preprint (1967), Brandeis University.Google Scholar
Cartan, H., Algèbres d’Eilenberg-MacLane et homotopie, Séminaire Henri Cartan, vol. 7 (Secrétariat mathématique, Paris, 1954/55).Google Scholar
Chałupnik, M., Koszul duality and extensions of exponential functors, Adv. Math. 218 (2008), 969–982.Google Scholar
Cline, E., Parshall, B. and Scott, L., Integral and graded quasi-hereditary algebras. I, J. Algebra 131 (1990), 126–160.Google Scholar
Day, B., Construction of biclosed categories, PhD thesis, University of New South Wales (1970).Google Scholar
Donkin, S., On Schur algebras and related algebras. I, J. Algebra 104 (1986), 310–328.Google Scholar
Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60.Google Scholar
Eilenberg, S. and MacLane, S., On the groups $H(\Pi , n). $ II. Methods of computation, Ann. of Math. (2) 60 (1954), 49–139.Google Scholar
Erdmann, K., Schur algebras of finite type, Q. J. Math. 44 (1993), 17–41.Google Scholar
Friedlander, E. M. and Suslin, A., Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209–270.Google Scholar
Green, J. A., Polynomial representations of GLn, Lecture Notes in Mathematics, vol. 830 (Springer, Berlin, 1980).Google Scholar
Im, G. B. and Kelly, G. M., A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), 75–88.Google Scholar
Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63–102.Google Scholar
Kuhn, N. J., Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (1994), 327–360.Google Scholar
Kuhn, N. J., Rational cohomology and cohomological stability in generic representation theory, Amer. J. Math. 120 (1998), 1317–1341.Google Scholar
Lewis, L. G. Jr., May, J. P. and Steinberger, M., Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213 (Springer, Berlin, 1986).Google Scholar
Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition (Oxford University Press, New York, 1995).Google Scholar
Mac Lane, S., Homology, Classics in Mathematics, reprint of the 1975 edition (Springer, Berlin, 1995).Google Scholar
Pirashvili, T., Introduction to functor homology, in Rational representations, the Steenrod algebra and functor homology, Panoramas et Synthèses, vol. 16 (Société Mathématique de France, Paris, 2003), 1–26.Google Scholar
Quillen, D., Higher algebraic K-theory. I, in Algebraic K-theory, I: higher K-theories, Seattle, WA, 1972, Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), 85–147.Google Scholar
Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295–366.Google Scholar
Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223.Google Scholar
Roby, N., Lois polynomes et lois formelles en théorie des modules, Ann. Sci. Éc. Norm. Supér. (3) 80 (1963), 213–348.Google Scholar
Schur, I., Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Dissertation, Universität Berlin (1901) (in I. Schur, Gesammelte Abhandlungen I (Springer, Berlin, 1973), 1–70).Google Scholar
Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121–154.Google Scholar
Totaro, B., Projective resolutions of representations of GL(n), J. Reine Angew. Math. 482 (1997), 1–13.Google Scholar
Touzé, A., Bar complexes and extensions of classical exponential functors, Preprint (2010), math.RT/1012.2724v2.Google Scholar
Touzé, A., Koszul duality and derivatives of non-additive functors, J. Pure Appl. Algebra, to appear, math.RT/1103.4580v1.Google Scholar
Verdier, J.-L., Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996).Google Scholar