Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture

Koenig, Steffen; Liu, Yuming; Zhou, Guodong

doi:10.1090/S0002-9947-2011-05358-4

Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture
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by Steffen Koenig, Yuming Liu and Guodong Zhou PDF

Trans. Amer. Math. Soc. 364 (2012), 195-232 Request permission

Abstract:

Derived equivalences and stable equivalences of Morita type, and new (candidate) invariants thereof, between symmetric algebras will be investigated, using transfer maps as a tool. Close relationships will be established between the new invariants and the validity of the Auslander–Reiten conjecture, which states the invariance of the number of non-projective simple modules under stable equivalence. More precisely, the validity of this conjecture for a given pair of algebras, which are stably equivalent of Morita type, will be characterized in terms of data refining Hochschild homology (via Külshammer ideals) being invariant and also in terms of cyclic homology being invariant. Thus, validity of the Auslander–Reiten conjecture implies a whole set of ring theoretic and cohomological data to be invariant under stable equivalence of Morita type, and hence also under derived equivalence. We shall also prove that the Batalin–Vilkovisky algebra structure of Hochschild cohomology for symmetric algebras is preserved by derived equivalence. The main tools to be developed and used are transfer maps and their properties, in particular a crucial compatibility condition between transfer maps in Hochschild homology and Hochschild cohomology via the duality between them.

References

Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
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, DOI 10.1017/CBO9780511623608
Christine Bessenrodt, Thorsten Holm, and Alexander Zimmermann, Generalized Reynolds ideals for non-symmetric algebras, J. Algebra 312 (2007), no. 2, 985–994. MR 2333196, DOI 10.1016/j.jalgebra.2007.02.028
S. Bouc, Bimodules, trace généralisée, et transfers en homologie de Hochschild. Preprint, 1997. Available on http://people.math.jussieu.fr/$\tilde {\phantom {x}}$bouc/
Thomas Breuer, László Héthelyi, Erzsébet Horváth, Burkhard Külshammer, and John Murray, Cartan invariants and central ideals of group algebras, J. Algebra 296 (2006), no. 1, 177–195. MR 2192603, DOI 10.1016/j.jalgebra.2005.05.003
Michel Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181-182 (1990), 61–92 (French). MR 1051243
Michel Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR 1308978, DOI 10.1007/978-94-017-1556-0_{1}
Daniel Dugger and Brooke Shipley, $K$-theory and derived equivalences, Duke Math. J. 124 (2004), no. 3, 587–617. MR 2085176, DOI 10.1215/S0012-7094-04-12435-2
Alex S. Dugas and Roberto Martínez-Villa, A note on stable equivalences of Morita type, J. Pure Appl. Algebra 208 (2007), no. 2, 421–433. MR 2277684, DOI 10.1016/j.jpaa.2006.01.007
Ching-Hwa Eu and Travis Schedler, Calabi-Yau Frobenius algebras, J. Algebra 321 (2009), no. 3, 774–815. MR 2488552, DOI 10.1016/j.jalgebra.2008.11.003
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
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
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
C. Kassel, Homology and cohomology of associative algebras - A concise introduction to cyclic homology, Notes of a course given in the Advanced School on Non-commutative Geometry at ICTP, Trieste in August 2004. Available at http://www-irma.u-strasbg.fr/~kassel/pubICTP04.html
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406, DOI 10.24033/asens.1689
Bernhard Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), no. 1-3, 223–273. MR 1492902, DOI 10.1016/S0022-4049(96)00085-0
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, DOI 10.1007/BFb0096366
B. Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, II, III, IV, J. Algebra 72 (1981), 1-7; J. Algebra 75 (1982), 59-69; J. Algebra 88 (1984), 279-291; J. Algebra 93 (1985), 310-323.
Burkhard Külshammer, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 425–442. MR 1112173, DOI 10.1007/978-3-0348-8658-1_{1}9
Markus Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebr. Represent. Theory 2 (1999), no. 2, 107–135. MR 1702272, DOI 10.1023/A:1009979222100
Yuming Liu, On stable equivalences induced by exact functors, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1605–1613. MR 2204270, DOI 10.1090/S0002-9939-05-08157-8
Yuming Liu, Summands of stable equivalences of Morita type, Comm. Algebra 36 (2008), no. 10, 3778–3782. MR 2458406, DOI 10.1080/00927870802160560
Yuming Liu and Changchang Xi, Constructions of stable equivalences of Morita type for finite dimensional algebras. II, Math. Z. 251 (2005), no. 1, 21–39. MR 2176462, DOI 10.1007/s00209-005-0786-9
Y. M. Liu, G. Zhou and A. Zimmermann Higman ideal, stable Hochschild homology and Auslander–Reiten conjecture, to appear in Math. Zeit., available at http://www.mathinfo.u-picardie.fr/alex/alexpapers.html.
Jean-Louis Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246, DOI 10.1007/978-3-662-11389-9
Luc Menichi, Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, $K$-Theory 32 (2004), no. 3, 231–251. MR 2114167, DOI 10.1007/s10977-004-0480-4
Zygmunt Pogorzały, Invariance of Hochschild cohomology algebras under stable equivalences of Morita type, J. Math. Soc. Japan 53 (2001), no. 4, 913–918. MR 1852888, DOI 10.2969/jmsj/05340913
Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. MR 1027750, DOI 10.1016/0022-4049(89)90081-9
Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
Jeremy Rickard, Some recent advances in modular representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 157–178. MR 1648606
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Thomas Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2351–2379 (English, with English and French summaries). MR 2498354, DOI 10.5802/aif.2417
Alexander Zimmermann, Invariance of generalised Reynolds ideals under derived equivalences, Math. Proc. R. Ir. Acad. 107 (2007), no. 1, 1–9. MR 2289795, DOI 10.3318/PRIA.2007.107.1.1
Alexander Zimmermann, Fine Hochschild invariants of derived categories for symmetric algebras, J. Algebra 308 (2007), no. 1, 350–367. MR 2290926, DOI 10.1016/j.jalgebra.2006.08.021
Alexander Zimmermann, Fine Hochschild invariants of derived categories for symmetric algebras, J. Algebra 308 (2007), no. 1, 350–367. MR 2290926, DOI 10.1016/j.jalgebra.2006.08.021