General linear and functor cohomology over finite fields

Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko and Andrei Suslin

Review from Zentralblatt MATH:

This paper contains many definite results concerning the cohomology of families of representations of $\text{GL}_n$ over a finite field $\bbfF_q$. A representation like the module $\wedge^2 \bbfF_q^n$ for $\text{GL}_n(\bbfF_q)$ makes sense for every $n$. Such a representation `for all $n$ simultaneously' is an object of the category $\Cal F$, or $\Cal F(\bbfF_q)$, of all functors from finite-dimensional $\bbfF_q$ vector spaces to $\bbfF_q$ vector spaces, studied extensively by Franjou, Lannes and Schwarz. Similarly we have the category $\Cal P$, or $\Cal P(\bbfF_q)$, of `strict polynomial functors' introduced by {\it E. M. Friedlander} and {\it A. Suslin} [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0918.20035)]. But the results of the authors are much stronger and much more explicit than in these works.

Reviewed by Wilberd van der Kallen

Keywords: polynomial functors; rational modules; algebraic groups; categories of functors; exponential functors; cohomology of representations; linear algebraic groups over finite fields; polynomial representations; cohomology rings; divided powers; symmetric powers; Ext groups

Classification (MSC2000): 20G05 20G10 18G05 18A22 20J05 20G40 14L15 18G15

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