  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, 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.
However, finite presheaves remain imperfectly understood, as illustrated by
examples. One important class of examples is shown to be provided by unstable
algebras of finite transcendence degree (under a necessary weak finiteness
condition).
  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 is
polynomial. This leads to a description of unstable Hopf algebras modulo
nilpotents in the spirit of Henn, Lannes and Schwartz.
