The degree filtration on polynomial functors and unstable modules

Lionel Schwartz (Paris/France)

One considers on one side the category of functors from the category of finite dimensional vector spaces over a finite prime field to the category of vector spaces. For simplicity of notations we will restrict to the caracteristic 2. The morphims are the natural transformations. This category contains as a full subcategory the category of polynomials functors in the sense of Eilenberg and Mac Lane.

On the other side one considers the category U of unstable modules over the Steenrod algebra (again for simplicity we will consider the case where the caracteristic is 2). The singular mod 2 cohomology of a space is such a module. One will introduce the notion of nilpotent module, and the quotient category U/N /il. We will explain briefly topological motivations for this study, in particular state Lannes’ theorem on maps from the classifying space of an elementary abelien p-group.

Then, we compare the filtration induced by the Eilenberg-MacLane degree on the category of polynomial functors, on one side, wi th on the other side, the weight filtration on the category of reduced unstable modules over the Steenrod algebra. Or, what is better on the (abelian) quotient category U/N il. One will show that they are in strict correspondance.

A reduced unstable module is said to be of weight less than n if it is trivial in degrees d such that α(d) > n.

In particular we will show how to associate to a reduced unstable module (or to an object in U/N il) a modular representation of the symmetric group S_{n} for any n. We will derive a brief description of simple objects in both settings, i.e. the one of polynomial functors and the one of unstable modules. The techniques involved there are quite close from group-representations theoretic ones.

As an application of what precedes we will discuss the classification of indecomposable injective objects in the category of polynomial functors, or in U/N il. Then, we will Krull-Schmidt decomposition of Carlsson’s unstable modules. This is equivalent to the description of the Krull-Schmidt decomposition of the functors S(i) which are direct limits under the square maps of the symmetric powers S^{2^{h}i} (see N. Kuhn’s talks).