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Polynomial functors over finite fields. (English) Zbl 0999.19003
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 369-388, Exp. No. 877 (2002).
This paper gives a review of some of the advances made during the 1990’s towards the study of polynomial functors over finite fields.
Let $$K$$ be a field, $$\mathcal V$$ be the category of finite-dimensional $$K$$-vector spaces and $$\mathcal F$$ the category of functors from $$\mathcal V$$ into the category of $$K$$-vector spaces. When $$K$$ is a prime field $$\mathcal F$$ is closely related to the category of unstable modules over the Steenrod algebra by work of H.-W. Henn, J. Lannes and L. Schwartz [Am. J. Math. 115, No. 5, 1053–1106 (1993; Zbl 0805.55011)].
Using cross-effects, there is a notion of polynomial functors in $$\mathcal F$$. There is an alternative called strict polynomial functors due to E. M. Friedlander and A. Suslin [Invent. Math. 127, No. 2, 209–270 (1997; Zbl 0945.14028)] related to the theory of polynomial representations of the general linear group.
The author describes some properties of $$\mathcal F$$ and the category $${\mathcal P}$$ of strict polynomial functors. The category $$\mathcal P$$ is more convenient for many purposes, and the author sketches the proof of the theorem of V. Franjou, E. M. Friedlander, A. Scorichenko and A. Suslin [Ann. Math. (2) 150, No. 2, 663–728 (1999; Zbl 0952.20035)] giving a close relationship between the Ext-groups over $$\mathcal P$$ and $$\mathcal F$$.
The last section contains a sketch of Suslin’s argument giving an isomorphism from $$\text{Ext}_{\mathcal F}^*(F,T)$$ of strict homogeneous functors $$F$$ and $$T$$ to the cohomology of $$\text{GL}(K)$$ with coefficients in $$\operatorname{Hom}_K(F,T)$$ whenever $$K$$ is a finite field.
For the entire collection see [Zbl 0981.00011].

##### MSC:
 19D55 $$K$$-theory and homology; cyclic homology and cohomology 55S10 Steenrod algebra 16G10 Representations of associative Artinian rings
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