Some finiteness results in the category U

This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups

La filtration de Krull de la categorie U et la cohomologie des espaces

Paper written in French -- English abstract: This paper proves a particular case of a conjecture of N. Kuhn. This conjecture is as follows. Consider the Gabriel-Krull filtration of the category U of unstable modules. Let U_n, n>=0, be the n-th step of this filtration. The category U is the smallest thick sub-category that contains all sub-categories U_n and is stable under colimit [L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics Series (1994)]. The category U_0 is the one of locally finite modules, i.e. the modules that are direct limit of finite modules. The conjecture is as follows, let X be a space then : * either H^*X is locally finite, * or H^*X does not belong to U_n, for all n. As an example the cohomology of a finite space, or of the loop space of a finite space are always locally finite. On the other side the cohomology of the classifying space of a finite group whose order is divisible by 2 does belong to any sub-category U_n. One proves this conjecture, modulo the additional hypothesis that all quotients of the nilpotent filtration are finitely generated. This condition is used when applying N. Kuhn's reduction of the problem. It is necessary to do it to be allowed to apply Lannes' theorem on the cohomology of mapping spaces.[N. Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. 141 (1995) 321-347].Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-27.abs.htm

Around conjectures of N. Kuhn

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod $p$ singular cohomology of a space, for $p$ a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes' mapping space theorem. The second (for the prime $2$) is based on an analysis of the $-1$ and $-2$ columns of the Eilenberg-Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between \Fp-vector spaces. The second result in particular exhibits the power of the functorial approach

Questions and conjectures about the modular representation theory of the general linear group GLn(F2) and the Poincar\'e series of unstable modules

This note is devoted to some questions about the representation theory over the finite field $\mathbb{F}_2$ of the general linear groups $\mathbb{GL_n(F_2)}$ and Poincar\'e series of unstable modules. The first draft was describing two conjectures. They were presented during talks made at VIASM in summer 2013. Since then one conjecture has been disproved, the other one has been proved. These results naturally lead to new questions which are going to be discussed. In winter 2013, Nguyen Dang Ho Hai proved the second conjecture, he disproved the first one in spring 2014. Up to now, the proof of the second one depends on a major topological result: the Segal conjecture. This discussion could be extended to an odd prime, but we will not do it here, just a small number of remarks will be made

Some finiteness results in the category U

This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups

Realizing a complex of unstable modules

6 pagesIn a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra. Content

LANNES' T FUNCTOR ON INJECTIVE UNSTABLE MODULES AND HARISH-CHANDRA RESTRICTION

In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) ∼ = L P ⊕ H * V 1 ⊗ L δ(P) , defining a functor δ from F p [M n (F p)]-projectives to F p [M n−1 (F p)]-projectives. We relate this new functor δ to classical constructions in the representation theory of the general linear groups

La fonction de partition de Minc et une conjecture de Segal pour certains spectres de Thom

On construit dans cet article une résolution injective minimale dans la catégorie \U des modules instables sur l'algèbre de Steenrod modulo $2$, de la cohomologie de certains spectres obtenus à partir de l'espace de Thom du fibré, associé à la représentation régulière réduite du groupe abélien élémentaire $(\Z/2)^n$, au dessus de l'espace $B(\Z/2)^n$. Les termes de la résolution sont des produits tensoriels de modules de Brown-Gitler $J(k)$ et de modules de Steinberg $L_n$ introduits par S. Mitchell et S. Priddy. Ces modules sont injectifs d'après J. Lannes et S. Zarati, de plus ils sont indécomposables. L'existence de cette résolution avait été conjecturée par Jean Lannes et le deuxième auteur. La principale indication soutenant cette conjecture était un résultat combinatoire de G. Andrews : la somme alternée des séries de Poincaré des modules considérées est nulle