Derived invariance of support varieties

Külshammer, Julian; Psaroudakis, Chrysostomos; Skartsæterhagen, Øystein

doi:10.1090/proc/13302

Derived invariance of support varieties
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by Julian Külshammer, Chrysostomos Psaroudakis and Øystein Skartsæterhagen

Proc. Amer. Math. Soc. 147 (2019), 1-14

DOI: https://doi.org/10.1090/proc/13302

Published electronically: October 22, 2018

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Abstract:

The (Fg) condition on Hochschild cohomology as well as the support variety theory are shown to be invariant under derived equivalence.

References

Maurice Auslander and Idun Reiten, Homologically finite subcategories, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 1–42. MR 1211476
Aslak Bakke Buan, Henning Krause, Nicole Snashall, and Øyvind Solberg, Axiomatic approach to support varieties, preprint in preparation, 2015.
Petter Andreas Bergh and Steffen Oppermann, Cohomology of twisted tensor products, J. Algebra 320 (2008), no. 8, 3327–3338. MR 2450729, DOI 10.1016/j.jalgebra.2008.08.005
Aslak Bakke Buan and Øyvind Solberg, Relative cotilting theory and almost complete cotilting modules, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 77–92. MR 1648616, DOI 10.1007/s10468-005-0341-8
Jon F. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), no. 1, 104–143. MR 723070, DOI 10.1016/0021-8693(83)90121-7
Xiao-Wu Chen and Yu Ye, Retractions and Gorenstein homological properties, Algebr. Represent. Theory 17 (2014), no. 3, 713–733. MR 3254767, DOI 10.1007/s10468-013-9415-1
Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
Karin Erdmann and Øyvind Solberg, Radical cube zero weakly symmetric algebras and support varieties, J. Pure Appl. Algebra 215 (2011), no. 2, 185–200. MR 2720683, DOI 10.1016/j.jpaa.2010.04.012
Rolf Farnsteiner, Tameness and complexity of finite group schemes, Bull. Lond. Math. Soc. 39 (2007), no. 1, 63–70. MR 2303520, DOI 10.1112/blms/bdl006
Eric M. Friedlander and Julia Pevtsova, Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc. 363 (2011), no. 11, 6007–6061. MR 2817418, DOI 10.1090/S0002-9947-2011-05330-4
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, On Gorenstein algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 389–404. MR 1112170, DOI 10.1007/978-3-0348-8658-1_{1}6
Thorsten Holm, Hochschild cohomology rings of algebras $k[X]/(f)$, Beiträge Algebra Geom. 41 (2000), no. 1, 291–301. MR 1745598
Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
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
Markus Linckelmann, Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero, Bull. Lond. Math. Soc. 43 (2011), no. 5, 871–885. MR 2854558, DOI 10.1112/blms/bdr024
Hiroshi Nagase, Hochschild cohomology and Gorenstein Nakayama algebras, Proceedings of the 43rd Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Soja, 2011, pp. 37–41. MR 2808393
Chrysostomos Psaroudakis, Øystein Skartsæterhagen, and Øyvind Solberg, Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements, Trans. Amer. Math. Soc. Ser. B 1 (2014), 45–95. MR 3274657, DOI 10.1090/S2330-0000-2014-00004-6
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 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
Claus Michael Ringel, The Gorenstein projective modules for the Nakayama algebras. I, J. Algebra 385 (2013), 241–261. MR 3049570, DOI 10.1016/j.jalgebra.2013.03.014
Joseph J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York, 2009. MR 2455920, DOI 10.1007/b98977
Øystein Skartsæterhagen, Singular equivalence and the (Fg) condition, J. Algebra 452 (2016), 66–93. MR 3461056, DOI 10.1016/j.jalgebra.2015.12.012
Øyvind Solberg, Support varieties for modules and complexes, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 239–270. MR 2258047, DOI 10.1090/conm/406/07659
Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR 2044054, DOI 10.1112/S002461150301459X
Zhengfang Wang, Singular equivalence of Morita type with level, J. Algebra 439 (2015), 245–269. MR 3373371, DOI 10.1016/j.jalgebra.2015.05.012
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
Alexander Zimmermann, Representation theory, Algebra and Applications, vol. 19, Springer, Cham, 2014. A homological algebra point of view. MR 3289041, DOI 10.1007/978-3-319-07968-4