Finite presheaves and $\protect \mathscr{A}$-finite generation of unstable algebras mod nilpotents

Abstract

Inspired by the work of Henn, Lannes and Schwartz on unstable algebras over the Steenrod algebra modulo nilpotents, a characterization of unstable algebras that are 𝒜-finitely generated up to nilpotents is given in terms of the associated presheaf. We do this by introducing the notion of a finite presheaf. In particular, this gives the natural characterization of the (co)analytic presheaves that are important in the theory of Henn, Lannes and Schwartz. An important source of examples is provided by unstable algebras of finite transcendence degree.