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Transactions of the American Mathematical Society

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Reduced standard modules and cohomology


Authors: Edward T. Cline, Brian J. Parshall and Leonard L. Scott
Journal: Trans. Amer. Math. Soc. 361 (2009), 5223-5261
MSC (2000): Primary 20Gxx
Published electronically: May 14, 2009
MathSciNet review: 2515810
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Abstract: First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper (1995). Internal to group theory, $ 1$-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985). One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology $ H^1_{\operatorname{gen}}(G,L) :=\underset{q\to\infty}{\lim} H^1(G(q),L)$ (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group $ G(q)$ of Lie type, with absolutely irreducible coefficients $ L$ (in the defining characteristic of $ G$), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on $ H^1(G(q),L)$ itself, still depending only on the root system. The generic $ H^1$ result, and related results for $ \operatorname{Ext}^1$, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules $ \Delta^{\text{\rm red}}(\lambda), \nabla_{\text{\rm red}}(\lambda)$, indexed by dominant weights $ \lambda$, for a reductive group $ G$. The modules $ \Delta^{\text{\rm red}}(\lambda)$ and $ \nabla_{\text{\rm red}}(\lambda)$ arise naturally from irreducible representations of the quantum enveloping algebra $ U_\zeta$ (of the same type as $ G$) at a $ p$th root of unity, where $ p>0$ is the characteristic of the defining field for $ G$. Finally, we apply our $ \operatorname{Ext}^1$-bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on $ H^1(G(q),L)$.


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Additional Information

Edward T. Cline
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: ecline@math.ou.edu

Brian J. Parshall
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: bjp8w@virginia.edu

Leonard L. Scott
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: lls2l@virginia.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04633-9
Received by editor(s): July 5, 2007
Received by editor(s) in revised form: September 5, 2007
Published electronically: May 14, 2009
Additional Notes: This research was supported in part by the National Science Foundation
Dedicated: We dedicate this paper to Toshiaki Shoji on the occasion of his 60th birthday
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.