Oberwolfach Seminars Volume 43 David J. Benson Srikanth Iyengar Henning Krause Representations of Finite Groups: Local Cohomology and Support David J. Benson Srikanth Iyengar Institute of Mathematics Department of Mathematics University of Aberdeen University of Nebraska Fraser Noble Building Lincoln, NE 68588-0130 King’s College USA Aberdeen AB24 3UE Scotland, UK Henning Krause Fakultät für Mathematik Universität Bielefeld P.O. Box 10 01 31 33501 Bielefeld Germany ISBN978-3-0348-0259-8 e-ISBN978-3-0348-0260-4 DOI10.1007/978-3-0348-0260-4 Springer Basel Dordrecht Heidelberg London New York LibraryofCongressControlNumber:2011941496 Mathematics Subject Classification (2010): 20J06, 13D99, 16E45 © Springer Basel AG 2012 This work is subject to copyright. 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Printedonacid-freepaper Springer Basel AG is part of Springer Science + Business Media (www.birkhauser-science.com) Contents Preface vii 1 Monday 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Modules over group algebras . . . . . . . . . . . . . . . . . . . . . 9 1.3 Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Tuesday 27 2.1 Perfect complexes over commutative rings . . . . . . . . . . . . . . 27 2.2 Brown representability and localisation . . . . . . . . . . . . . . . . 33 2.3 The stable module category of a finite group . . . . . . . . . . . . 36 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Wednesday 47 3.1 Local cohomology and support . . . . . . . . . . . . . . . . . . . . 47 3.2 Koszul objects and support . . . . . . . . . . . . . . . . . . . . . . 50 3.3 The homotopy category of injectives . . . . . . . . . . . . . . . . . 54 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Thursday 63 4.1 Stratifying triangulated categories . . . . . . . . . . . . . . . . . . 63 4.2 Consequences of stratification . . . . . . . . . . . . . . . . . . . . . 69 4.3 The Klein four group . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5 Friday 79 5.1 Localising subcategories of D(A) . . . . . . . . . . . . . . . . . . . 79 5.2 Elementary abelian 2-groups . . . . . . . . . . . . . . . . . . . . . 82 5.3 Stratification for arbitrary finite groups . . . . . . . . . . . . . . . 86 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 vi Contents A Support for modules over commutative rings 93 A.1 Big support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.2 Serre subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 A.3 Localising subcategories . . . . . . . . . . . . . . . . . . . . . . . . 95 A.4 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 A.5 Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 A.6 Specialization closed sets. . . . . . . . . . . . . . . . . . . . . . . . 97 Bibliography 99 Index 103 Preface These are the notes from an Oberwolfach Seminar which we ran from 23–29 May 2010. There were 24 graduate student and postdoctoral participants. Each morn- ing consisted of three lectures, one from each of the organisers. The afternoons consisted of problem sessions, apart from Wednesday which was reserved for the traditional hike to St. Roman. We have tried to be reasonably faithful to the lec- tures and problem sessions in these notes, and have added only a small amount of new material for clarification. The seminar focused on recent developments in classification methods in commutative algebra, group representation theory and algebraic topology. These methods were initiated by Hopkins back in 1987 [35], with the classification of the thick subcategories of the derived category of bounded complexes of finitely generated projective modules over a commutative noetherian ring R, in terms of specialisation closed subsets of SpecR. Neeman [44] (1992) clarified Hopkins’ theorem and used analogous methods to classify the localising subcategories of the derived category of unbounded complexes of modules in terms of arbitrary subsets of SpecR. In 1997, Benson, Carlson and Rickard [9] proved the thick sub- category theorem for modular representation theory of a finite p-group G over an algebraically closed field k of characteristic p. Namely, the thick subcategories of the stable category of finitely generated kG-modules are classified by the special- isation closed subsets of the homogeneous non-maximal prime ideals in H∗(G,k), the cohomology ring. The corresponding theorem for the localising subcategories of the stable category of all kG-modules has only recently been achieved, in the paper [11] by the three organisers of the seminar. Thick subcategories Localising subcategories of compact objects of all objects D(R) Hopkins 1987 Neeman 1992 StMod(kG) Benson, Carlson Benson, Iyengar and Rickard 1997 and Krause 2008 viii Preface Intheprocessofachievingtheclassificationofthelocalisingsubcategoriesof StMod(kG), a general machinery was established for such classification theorems in a triangulated category; see [10, 12]. It is also worth mentioning at this stage the work of Hovey, Palmieri and Strickland [36], who did a great deal to clarify the appropriate settings for these theorems. The general setup involves a graded commutative noetherian ring R acting on a compactly generated triangulated category with small coproducts T. Write SpecR for the set of homogeneous prime ideals of R. For each p∈SpecR there is a local cohomology functor Γp: T → T. The support of an object X is defined to be the subset of SpecR consisting of those p such that ΓpX is non-zero. The object of the game is to establish conditions under which this notion of support classifies the localising subcategories of T. This is given in terms of two conditions. The first is the local-global principle that says for each object X in T, the localising subcategory of T generated by X is the same as that generated by {ΓpX}asprunsovertheelementsofSpecR.Thesecondisaminimalitycondition, which requires that each ΓpT is either a minimal localising subcategory of T or it is zero. Under these two conditions, we say that T is stratified by the action of R, and then we obtain a classification theorem. In the case of the derived category D(R), Neeman’s classification made es- sential use of the existence of “field objects” – for a prime ideal p of R, the field object is the complex consisting of the field of fractions of R/p, concentrated in a single degree. One of the principle obstructions to carrying out the classification inthefinitegroupcaseisalackoffieldobjects;theobstructiontheoryofBenson, Krause and Schwede [15, 16] can be used to show that the required field objects usually do not exist. Circumventing this involves an elaborate series of changes of category, and machinery for transferring stratification along such changes of cat- egory. For a general finite group, the strategy is first to use Quillen stratification to reduce to elementary abelian p-groups, where there are still not enough field objects, but then to use a Koszul construction to reduce to an exterior algebra for which there are enough field objects. At this stage, a version of the Bernstein– Gelfand–Gelfand correspondence can be used to get to a graded polynomial ring, where the problem is solved. One consequence of this strategy is that we obtain classification theorems in a number of situations along the route. In these notes we manage to give a complete proof in the case of charac- teristic two, where matters are considerably simplified by the fact that the group algebraof an elementary abelian 2-group is already an exterior algebra. We found it frustrating that in spite of having an entire week of lectures to explain the theory, we were not able to give a complete proof of the classification theorem for localising subcategories of StMod(kG), in odd characteristic. An overview of the classification in general characteristic is given in Section 3.3, while the proof in characteristic two may be obtained by combining Theorems 5.4 and 5.19 with results from Section 3.3. Preface ix A guide to these notes In this volume, we have attempted to stick as closely as possible to the format of the Oberwolfach seminar. So the notes are divided into five chapters with four sections in each, corresponding to the five days with three lectures each morning and a problem session in the afternoon. The lecturing, and writing, styles of the three authors are different, and we have not tried to alter that for the purpose of thesenotes.Inparticular,thereisasmallamountofrepetition.Butwehavetried tobeconsistentaboutimportantdetailssuchasnotation,andgradingeverything cohomologically rather than homologically. Prerequisites for this seminar consist of a solid background in algebra, in- cluding the basic theory of rings and modules, Artin–Wedderburn theory, Krull– Remak–Schmidttheorem;basiccommutativealgebrafromthefirstchaptersofthe book of Atiyah and MacDonald; and basic homological algebra including derived functors,ExtandTor.Theappendix,describingthetheoryofsupportformodules overacommutativering,isalsonecessarybackgroundmaterialfromcommutative algebra that is not easy to find in the literature in the exact form in which we require it. The following books may be helpful. [1] M. Atiyah and I. MacDonald, Commutative Algebra. Addison-Wesley, 1969. [2] D. J. Benson, Representations and cohomology of finite groups I, II, Cam- bridgeStudiesinAdvancedMathematics30,31.CambridgeUniversityPress, 2nd edition, 1998. [3] W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Studies in Ad- vanced Mathematics 39. Cambridge University Press, 2nd edition, 1998. [4] R.Hartshorne,Localcohomology:AseminargivenbyA.Grothendieck(Har- vard, 1961), Lecture Notes in Math. 41. Springer-Verlag, 1967. [5] A. Neeman, Triangulated categories, Annals of Mathematics Studies 148. Princeton University Press, 2001. [6] C.Weibel,Homologicalalgebra,CambridgeStudiesinAdvancedMathemat- ics 38. Cambridge University Press, 1994. About the exercises: These are from the problem sessions conducted dur- ing the seminar, though we have added a few more. Some are routine verifica- tions/computations that have been omitted in the text, while others are quite substantial, and given with the implicit assumption (or hope) that, if necessary, readers would hunt for solutions in other sources. Acknowledgments The first and second authors would like to thank the Humboldt Foundation for their generous support of the research that led to the seminar. The second author was also partly supported by NSF grant DMS 0903493. The three of us thank the