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].

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].

Reviewer: Bjørn Dundas (Trondheim)

##### MSC:

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |

55S10 | Steenrod algebra |

16G10 | Representations of associative Artinian rings |