Finite presheaves and \(\mathscr{A}\)-finite generation of unstable algebras mod nilpotents
type: Item
instance of: scholarly article
title: Finite presheaves and \(\mathscr{A}\)-finite generation of unstable algebras mod nilpotents
zbMATH Open document ID: 1444.55010
publication date: 2019-10-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1707.07468
review text: 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 monomor…
reviewed by: Rui Miguel Saramago
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55U99
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 7114111
zbMATH Keywords: unstable algebra
zbMATH Keywords: nilpotents
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: finite generation
zbMATH Keywords: unstable Hopf algebra
zbMATH Keywords: presheaf
zbMATH Keywords: polynomial functor
author: Geoffrey Powell
MaRDI profile type: Publication
OpenAlex ID: W2974134443
arXiv ID: 1707.07468
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DOI: 10.5802/AIF.3292
published in: Annales de l’institut Fourier
description: scientific article