×

The Avrunin and Scott theorem and a truncated polynomial algebra. (English) Zbl 1102.16007

Let \(G\) be a finite group and \(k\) be an algebraically closed field of characteristic \(p>0\). Over the past 30 years, the cohomology and representation theory of \(kG\)-modules has been greatly influenced by the idea of cohomological support varieties which provide a geometric way to study cohomology. The success of this tool in the modular representation theory of finite groups has led to its use in many other areas. One key to this success has been the ability to provide non-cohomological descriptions of support varieties.
The first such description, in the case of finite groups, was conjectured by J. Carlson. More precisely, for an elementary Abelian \(p\)-group, Carlson introduced the notion of a rank variety for a module which is defined representation-theoretically without using cohomology. For the trivial module \(k\), the support variety and rank variety are naturally isomorphic. For any finite-dimensional module, the support (rank) variety is a subvariety of the support (rank) variety for the trivial module. Carlson conjectured that under this isomorphism, for any finite-dimensional module, the support variety and rank varieties should correspond. This conjecture was first proven by G. Avrunin and L. Scott and is the theorem referred to in the title of the paper.
The goal of this paper is to investigate the analogous question in the context of modules for a truncated polynomial algebra \(\Lambda_m=k[X_1,X_2,\dots,X_m]/(X_i^2)\) over an algebraically closed field \(k\) of arbitrary characteristic. For \(\Lambda_m\), support varieties are defined using Hochschild cohomology. This is based on work of N. Snashall and Ø. Solberg [Proc. Lond. Math. Soc. (3) 88, No. 3, 705–732 (2004; Zbl 1067.16010)] and of the authors with N. Snashall, Ø. Solberg and R. Taillefer [\(K\)-Theory 33, No. 1, 67–87 (2004; Zbl 1116.16007)]. Rank varieties were introduced by the authors [in Math. Z. 247, No. 3, 441–460 (2004; Zbl 1078.16008)].
The main result of this paper is to show that for any finite-dimensional \(\Lambda_m\)-module the support variety and rank variety again correspond under an isomorphism identifying the two varieties for the trivial module. The authors use an interesting new approach of identifying both the support and rank varieties in terms of stable maps. With this description, they show that the two varieties correspond. The authors also show that their methods provide a new proof of the Avrunin and Scott theorem for elementary Abelian groups.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S34 Group rings
20J06 Cohomology of groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avramov, L. L.; Buchweitz, R. O., Support varieties and cohomology over complete intersections, Invent. Math., 142, 285-318 (2000) · Zbl 0999.13008
[2] Avrunin, G. S.; Scott, L. L., Quillen stratification for modules, Invent. Math., 66, 277-286 (1982) · Zbl 0489.20042
[3] Bendel, C.; Friedlander, E.; Suslin, A., Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc., 10, 693-728 (1997) · Zbl 0960.14023
[4] Bendel, C.; Friedlander, E.; Suslin, A., Support varieties for infinitesimal group schemes, J. Amer. Math. Soc., 10, 729-759 (1997) · Zbl 0960.14024
[5] Bendel, C.; Nakano, D. K., Complexes and vanishing of cohomology for group schemes, J. Algebra, 214, 668-713 (1999) · Zbl 0942.16011
[6] Benson, D. J., Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Stud. Adv. Math., vol. 30 (1991), Cambridge Univ. Press, Reprinted in paperback, 1998 · Zbl 0718.20001
[7] Benson, D. J., Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge Stud. Adv. Math., vol. 31 (1991), Cambridge Univ. Press, reprinted in paperback, 1998 · Zbl 0731.20001
[8] Carlson, J. F., The varieties and cohomology ring of a module, J. Algebra, 85, 104-143 (1983) · Zbl 0526.20040
[9] Carlson, J. F., The variety of an indecomposable module is connected, Invent. Math., 77, 291-299 (1984) · Zbl 0543.20032
[10] Carlson, J. F., The cohomology ring of a module, J. Pure Appl. Algebra, 36, 105-121 (1985) · Zbl 0565.20003
[11] Dade, E. C., Endo-permutation modules over \(p\)-groups, II, Ann. of Math., 108, 317-346 (1978) · Zbl 0404.16003
[12] Erdmann, K.; Holloway, M., Rank varieties and projectivity for a class of local algebras, Math. Z., 247, 441-460 (2004) · Zbl 1078.16008
[13] Erdmann, K.; Holloway, M.; Snashall, N.; Solberg, Ø.; Taillefer, R., Support varieties for selfinjective algebras, \(K\)-Theory, 33, 67-87 (2004) · Zbl 1116.16007
[14] Friedlander, E. M.; Parshall, B. J., Support varieties for restricted Lie algebras, Invent. Math., 86, 553-562 (1986) · Zbl 0626.17010
[15] Friedlander, E. M.; Pevtsova, J., Representation theoretic support spaces for finite group schemes, Amer. J. Math., 127, 379-420 (2005) · Zbl 1072.20009
[16] Friedlander, E. M.; Suslin, A., Cohomology of finite group schemes over a field, Invent. Math., 127, 209-270 (1997) · Zbl 0945.14028
[17] Ginzburg, V.; Kumar, S., Cohomology of quantum groups at roots of unity, Duke Math. J., 69, 179-198 (1993) · Zbl 0774.17013
[18] Holm, T., Hochschild cohomology rings of algebras \(k [X] /(f)\), Beiträge Algebra Geom., 1, 291-301 (2000) · Zbl 0961.13006
[19] Jantzen, J. C., Kohomologie von \(p\)-Lie Algebren und nilpotente Elemente, Abh. Math. Sem. Univ. Hamburg, 76, 191-219 (1986) · Zbl 0614.17008
[20] Nakano, D. K.; Palmieri, J., Support varieties for the Steenrod algebra, Math. Z., 227, 663-684 (1998) · Zbl 0901.55008
[21] Ostrik, V., Cohomological supports for quantum groups, Funct. Anal. Appl., 32, 237-246 (1999), (translation from Russian) · Zbl 0981.17010
[22] Palmieri, J. H., Quillen stratification for the Steenrod algebra, Ann. of Math., 149, 421-449 (1999) · Zbl 0932.55021
[23] Parshall, B.; Wang, J. P., Cohomology of infnitesimal quantum groups, I, Tohoku Math. J., 44, 395-423 (1992) · Zbl 0770.17006
[24] Parshall, B.; Wang, J. P., Cohomology of infnitesimal quantum groups: The quantum dimension, Canad. J. Math., 45, 1276-1298 (1993) · Zbl 0835.17009
[25] Snashall, N.; Solberg, Ø., Support varieties and Hochschild cohomology rings, Proc. London Math. Soc., 88, 705-732 (2004) · Zbl 1067.16010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.