Finite presheaves and \(\mathscr{A}\)-finite generation of unstable algebras mod nilpotents
Given an unstable algebra \(K\) over the mod \(p\) Steenrod algebra \(\mathcal{A}\), Lannes's \(T\)-functor provides an associated presheaf \({\mathfrak g}K\) on the category of finite-dimensional \({\mathbb{F}}_p\)-vector spaces, given by \(V \mapsto \mathrm{Hom}_\mathcal{K}(K, H^*(BV;\mathbb{F}_p))\) (for each such vector space \(V\)), where \(\mathcal{K}\) denotes the category of unstable algebras over the mod \(p\) Steenrod algebra. For each \(d \in \mathbb{N}\), this construction provides an equivalence between the (opposite) category of unstable algebras of transcendence degree at most \(d\) modulo nilpotents and the category of profinite right \(\mathrm{End}(\mathbb{F}^d)\)-sets (Theorem 2.28). One can also develop a criterion for an unstable algebra to be Noetherian up to nilpotents in terms of its associated presheaf (Theorem 2.29). These results are due to \textit{H.-W. Henn} et al. [Am. J. Math. 115, No. 5, 1053--1106 (1993; Zbl 0805.55011)]. In the present paper, the author expands these results to the setting of unstable algebras that are \(\mathcal{A}\)-finitely generated up to nilpotents. For this, the notion of a finite presheaf is developed in Section 3, and the main result comes as Corollary 4.4: \(K\) is \(\mathcal{A}\)-finitely generated up to nilpotents if and only if its associated presheaf \({\mathfrak g}K\) is a finite presheaf. This leads to a new interpretation of some results by Henn, Lannes and Schwartz, in particular an equivalence between the (opposite) category of unstable algebras modulo nilpotents and the category of presheaves of profinite sets on \(\mathcal{V}_f\) that are coanalytic covariant functors, where \(\mathcal{V}_f\) denotes the category of finite-dimensional \(\mathbb{F}_p\)-vector spaces (Theorem 4.6). The definition of finite presheaf appears in Section 3.1: a presheaf of \({\mathbb{F}}_p\)-vector spaces is finite if it has a finite composition series, and a set-valued presheaf \(X\) is finite if there exists a monomorphism \(X \hookrightarrow F_X\) in the category of presheafs of sets in \(\mathcal{V}_f\), for some finite presheaf \(F_X\) of \({\mathbb{F}}_p\)-vector spaces. In some cases, the equivalence coming from Corollary 4.4 can be further refined. For example, if the unstable algebra \(K\) has finite transcendence degree, one gets Corollary 5.9: being \(\mathcal{A}\)-finitely generated up to nilpotents is equivalent to the associated presheaf taking values in finite sets. Section 6 studies conditions for determining if a presheaf that takes values in finite sets is indeed a finite presheaf. This is done in terms of the growth function for the presheaf \(X\), defined on \(\mathbb{N}\) by \(\gamma_X(t) = \mathrm{log}_p |X({\mathbb{F}}_p^t)|\), and the main result is Proposition 6.4, stating in particular that a necessary condition for a presheaf to be finite is for its associated \(\gamma_X\) to be of polynomial growth. Another particular case is that of connected Hopf algebras over \({\mathbb{F}}_p\) (whose underlying algebra is an unstable algebra over \(\mathcal{A}\)), which are the subject of the second part of this work, from Section 7 onwards. The main results along these lines appear in Theorems 8.7 and 8.12: as in Corollary 4.4, the underlying unstable algebra is \(\mathcal{A}\)-finitely generated up to nilpotents if and only if its associated presheaf \({\mathfrak g}K\) is a finite presheaf, but now one gets two additional equivalent conditions, namely that the growth function for this associated presheaf has polynomial growth, and also that \({\mathfrak g}K\) takes values in finite \(p\)-groups and defines a polynomial functor. This notion of polynomial functor (defined on some categories) is reviewed and discussed in a final appendix.
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