Presheaves on $\mathcal{VI}$, $nil$-closed unstable algebras and their centres

TL;DR

This paper explores the structure of nil-closed unstable algebras over the Steenrod algebra using presheaves on vector spaces, focusing on their centres and related algebraic structures, generalizing key theorems.

Contribution

It introduces a groupoid construction to determine the centre of nil-closed unstable algebras and generalizes the second theorem of Adams-Wilkerson.

Findings

Centre of algebra determined by associated groupoid

Generalization of Adams-Wilkerson theorem to sub-algebras

Primitive elements form a noetherian algebra

Abstract

A $nil$-closed, noetherian, unstable algebra $K$ over the Steenrod Algebra is determined, up to isomorphism, by the functor $\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$, which is a presheaf on the category $\mathcal{VI}$ of finite dimensional vector spaces and injections, by the theory of Henn-Lannes-Schwartz. In this article, we use this theory to study the centre, in the sense of Heard, of a $nil$-closed noetherian unstable algebra. For $F$ a presheaf on $\mathcal{VI}$, we construct a groupoid $\mathcal{G}_F$ which encodes $F$. Then, taking $F:=\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$, we show how the centre of $K$ is determined by the associated groupoid. We also give a generalisation of the second theorem of Adams-Wilkerson, defining sub-algebras $H^*(W)^\mathcal{G}$ of $H^*(W)$ for appropriate groupoids $\mathcal{G}$. There is a $H^*(C)$-comodule structure on $K$ that is associated with the centre. For $K$ integral, we explain how the algebra of primitive elements of this $H^*(C)$-comodule structure is also determined by the groupoid associated with $\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$. Along the way, we prove that this algebra of primitive elements is also noetherian.