Fibrewise stratification of group representations

. Given a finite cocommutative Hopf algebra A over a commutative regular ring R , the lattice of localising tensor ideals of the stable category of Gorenstein projective A -modules is described in terms of the corresponding lattices for the fibres of A over the spectrum of R . Under certain natural conditions on the cohomology of A over R , this yields a stratification of the stable category. These results apply when A is the group algebra over R of a finite group, and also when A is the exterior algebra on a finite free R -module.

[1]  H. Krause,et al.  A class of Gorenstein algebras and their dualities , 2023, 2303.04893.

[2]  J. Kujawa,et al.  Support varieties and modules of finite projective dimension for modular Lie superalgebras , 2019, Algebra & Number Theory.

[3]  R. Buchweitz.  Maximal Cohen–Macaulay Modules and Tate Cohomology , 2021, Mathematical Surveys and Monographs.

[4]  H. Krause,et al.  The Nakayama functor and its completion for Gorenstein algebras , 2020, Bulletin de la Société mathématique de France.

[5]  Paul Balmer,et al.  Grothendieck–Neeman duality and the Wirthmüller isomorphism , 2015, Compositio Mathematica.

[6]  H. Krause,et al.  Stratification for module categories of finite group schemes , 2015, 1510.06773.

[7]  H. Krause,et al.  Module categories for group algebras over commutative rings , 2012, 1208.1291.

[8]  Greg Stevenson.  Support theory via actions of tensor triangulated categories , 2011, 1105.4692.

[9]  H. Krause,et al.  Stratifying triangulated categories , 2009, 0910.0642.

[10]  H. Krause,et al.  Colocalizing subcategories and cosupport , 2010, 1008.3701.

[11]  Overtoun M. G. Jenda,et al.  Relative homological algebra , 1956 .

[12]  S. Iyengar,et al.  Homological Dimensions and Regular Rings , 2009, 0908.2470.

[13]  H. Krause,et al.  Stratifying modular representations of finite groups , 2008, 0810.1339.

[14]  H. Krause,et al.  Local cohomology and support for triangulated categories , 2007, math/0702610.

[15]  H. Krause,et al.  Acyclicity versus total acyclicity for complexes over noetherian rings , 2005, Documenta Mathematica.

[16]  H. Krause.  the stable derived category of a noetherian scheme , 2004, Compositio Mathematica.

[17]  B. Guillou,et al.  Algebraic K-theory , 2020, Oberwolfach Reports.

[18]  Peter Jørgensen.  The homotopy category of complexes of projective modules , 2003, math/0312088.

[19]  D. Orlov,et al.  Triangulated categories of singularities and D-branes in Landau-Ginzburg models , 2003, math/0302304.

[20]  A. Suslin,et al.  Cohomology of finite group schemes over a field , 1997 .

[21]  Mark Hovey,et al.  Axiomatic stable homotopy theory , 1997 .

[22]  D. Benson,et al.  Complexity and varieties for infinitely generated modules, II , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Amnon Neeman,et al.  The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , 1996 .

[24]  D. Benson,et al.  Complexity and varieties for infinitely generated modules , 1995, Mathematical Proceedings of the Cambridge Philosophical Society.

[25]  Edgar E. Enochs,et al.  On Cohen-Macaulay rings , 1994 .

[26]  A. Neeman,et al.  The chromatic tower for D(R) , 1992 .

[27]  A. Neeman.  The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel , 1992 .

[28]  N. Habegger,et al.  VARIETIES FOR MODULES AND A PROBLEM OF STEENROD , 1987 .

[29]  Robert Steinberg,et al.  Representations of Algebraic Groups , 1963, Nagoya Mathematical Journal.

[30]  Leonard Evens,et al.  The cohomology ring of a finite group , 1961 .

[31]  I. Kaplansky.  Modules over Dedekind rings and valuation rings , 1952 .