Polynomial Filtrations and Lannes' T Functor

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Abstract

This paper defines and studies the polynomial filtration [pk ˜Δ] of the shift functor ˜Δ: ℱ → ℱ, where ℱ is the category of functors between F-vector spaces over a finite field F. The functors [pk ˜Δ] correspond to a system of functors (pkT): 𝒰 → 𝒰, related to Lannes' T-functor on the category 𝒰 of unstable modules over the Steenrod algebra. The main results concern the behaviour of the quotients ˜∇s ≔ ˜Δ/[ps−1˜Δ] filtrations by ˜∇s-nilpotent functors are introduced and it is shown that the full subcategory of ˜∇s-nilpotent functors is thick.

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