Realizability of modules over Tate cohomology

Benson, David; Krause, Henning; Schwede, Stefan

doi:10.1090/S0002-9947-03-03373-7

Realizability of modules over Tate cohomology
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by David Benson, Henning Krause and Stefan Schwede PDF

Trans. Amer. Math. Soc. 356 (2004), 3621-3668 Request permission

Abstract:

Let $k$ be a field and let $G$ be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology $\gamma _G\in H\!H^{3,-1}\hat H^*(G,k)$ with the following property. Given a graded $\hat H^*(G,k)$-module $X$, the image of $\gamma _G$ in $\operatorname {Ext}^{3,-1}_{\hat H^*(G,k)}(X,X)$ vanishes if and only if $X$ is isomorphic to a direct summand of $\hat H^*(G,M)$ for some $kG$-module $M$. The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra $A$, there is also a canonical element of Hochschild cohomology $H\!H^{3,-1}H^*(A)$ which is a predecessor for these obstructions.

References

M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
Hans-Joachim Baues, On the cohomology of categories, universal Toda brackets and homotopy pairs, $K$-Theory 11 (1997), no. 3, 259–285. MR 1451757, DOI 10.1023/A:1007796409912
Hans Joachim Baues and Winfried Dreckmann, The cohomology of homotopy categories and the general linear group, $K$-Theory 3 (1989), no. 4, 307–338. MR 1047191, DOI 10.1007/BF00584524
Hans-Joachim Baues and Andy Tonks, On sum-normalised cohomology of categories, twisted homotopy pairs and universal Toda brackets, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 188, 405–433. MR 1460232, DOI 10.1093/qmath/47.4.405
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
D. J. Benson, Complexity and varieties for infinite groups. I, II, J. Algebra 193 (1997), no. 1, 260–287, 288–317. MR 1456576, DOI 10.1006/jabr.1996.6996
D. J. Benson, Complexity and varieties for infinite groups. I, II, J. Algebra 193 (1997), no. 1, 260–287, 288–317. MR 1456576, DOI 10.1006/jabr.1996.6996
D. J. Benson and Jon F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), no. 2, 107–129. MR 1182934, DOI 10.1016/0022-4049(92)90116-W
D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory. II, Algebr. Represent. Theory 4 (2001), no. 4, 395–404. MR 1863392, DOI 10.1023/A:1012475019810
A. J. Berrick and A. A. Davydov, Splitting of Gysin extensions, Algebr. Geom. Topol. 1 (2001), 743–762. MR 1875616, DOI 10.2140/agt.2001.1.743
Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234. MR 1214458
Jon F. Carlson, Modules and group algebras, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR 1393196, DOI 10.1007/978-3-0348-9189-9
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
D. Christensen, B. Keller, and A. Neeman, Failure of Brown representability in derived categories, Topology 40 (2001), 1339–1361.
Jonathan Cornick and Peter H. Kropholler, Homological finiteness conditions for modules over strongly group-graded rings, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 43–54. MR 1373347, DOI 10.1017/S030500410007465X
Jonathan Cornick and Peter H. Kropholler, Homological finiteness conditions for modules over group algebras, J. London Math. Soc. (2) 58 (1998), no. 1, 49–62. MR 1666074, DOI 10.1112/S0024610798005729
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144017
Peter Freyd, Stable homotopy, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 121–172. MR 0211399
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306, DOI 10.1007/978-3-662-03220-6
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
François Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), no. 1, 39–64 (French, with English summary). MR 1181092, DOI 10.1016/0022-4049(92)90009-5
Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167, DOI 10.1007/BF02564452
T. V. Kadeishvili, The algebraic structure in the homology of an $A(\infty )$-algebra, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), no. 2, 249–252 (1983) (Russian, with English and Georgian summaries). MR 720689
T. V. Kadeishvili, The structure of the $A(\infty )$-algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19–27 (Russian, with English summary). MR 1029003
M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. MR 939472, DOI 10.1007/BF01393744
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406, DOI 10.24033/asens.1689
—, Introduction to $A$-infinity algebras and modules, Homology, Homotopy and Applications 3 (2001), 1–35.
Peter H. Kropholler, Hierarchical decompositions, generalized Tate cohomology, and groups of type $(\textrm {FP})_\infty$, Combinatorial and geometric group theory (Edinburgh, 1993) London Math. Soc. Lecture Note Ser., vol. 204, Cambridge Univ. Press, Cambridge, 1995, pp. 190–216. MR 1320283
H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam, 1983. Modules over the Steenrod algebra and the stable homotopy category. MR 738973
Guido Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), no. 3, 293–300. MR 1269317, DOI 10.1016/0166-8641(94)90081-7
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
Jeremy Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), no. 1, 149–170. MR 1462832, DOI 10.1112/S0024610797005309
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
John Tate, Nilpotent quotient groups, Topology 3 (1964), no. suppl, suppl. 1, 109–111. MR 160822, DOI 10.1016/0040-9383(64)90008-4
Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR 1453167
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
George W. Whitehead, Recent advances in homotopy theory, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 5, American Mathematical Society, Providence, R.I., 1970. MR 0309097