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Inspired by the work of Henn, Lannes and Schwartz on unstable algebras overthe Steenrod algebra modulo nilpotents, a characterization of unstable algebrasthat are $A$-finitely generated up to nilpotents is given in terms of theassociated presheaf, by introducing the notion of a finite presheaf. Inparticular, this gives the natural characterization of the (co)analyticpresheaves that are important in the theory of Henn, Lannes and Schwartz.However, finite presheaves remain imperfectly understood, as illustrated byexamples. One important class of examples is shown to be provided by unstablealgebras of finite transcendence degree (under a necessary weak finitenesscondition).  For unstable Hopf algebras, it is shown that the situation is much better:the associated presheaf is finite if and only if its growth function ispolynomial. This leads to a description of unstable Hopf algebras modulonilpotents in the spirit of Henn, Lannes and Schwartz.

Finite presheaves and $A$-finite generation of unstable algebras mod nilpotents