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Polynomial filtrations and Lannes’ \(T\)-functor. (English) Zbl 0892.55009

The author’s abstract. “This paper defines and studies the polynomial filtration \([p_k \widetilde \Delta]\) of the shift functor \(\widetilde \Delta: {\mathcal F} \to {\mathcal F}\), where \({\mathcal F}\) is the category of functors between \(\mathbb{F}\)-vector spaces over a finite field \(\mathbb{F}\). The functors \([p_k \widetilde \Delta]\) correspond to a system of functors \((p_kT): {\mathcal U} \to {\mathcal U}\), related to Lannes’ \(T\)-functor on the category \({\mathcal U}\) of unstable modules over the Steenrod algebra. The main results concern the behaviour of the quotients \(\widetilde \nabla_s: =\widetilde \Delta/[p_{s-1} \widetilde \Delta]\); filtrations by \(\widetilde \nabla_s\)-nilpotent functors are introduced and it is shown that the full subcategory of \(\widetilde \nabla_s\)-nilpotent functors is thick.

MSC:

55S10 Steenrod algebra
20G05 Representation theory for linear algebraic groups
18G05 Projectives and injectives (category-theoretic aspects)
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