Polynomial Filtrations and Lannes' T Functor

This paper defines and studies the polynomial filtration [p_k] of the shift functor : F , where F is the category of functors between F-vector spaces over a finite field F. The functors [p_k] correspond to a system of functors (p_k T):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:=~/[p_s-1~ filtrations by ~_s-nilpotent functors are introduced and it is shown that the full subcategory of _s-nilpotent functors is thick.