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

Abstract

A $nil$-closed, noetherian, unstable algebra $K$ over the Steenrod Algebra is determined, up to isomorphism, by the functor ${\text{Hom}}_{\mathcal{K}\text{f.g.}}(K,H^{\ast }(\_))$, 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 $F:={\text{Hom}}_{\mathcal{K}\text{f.g.}}(K,H^{\ast }(\_))$, 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 $H^{\ast }(W{)}^{\mathcal{G}}$ of $H^{\ast }(W)$ for appropriate groupoids $\mathcal{G}$. There is a $H^{\ast }(C)$-comodule structure on $K$ that is associated with the centre. For $K$ integral, we explain how the algebra of primitive elements of this $H^{\ast }(C)$-comodule structure is also determined by the groupoid associated with ${\text{Hom}}_{\mathcal{K}\text{f.g.}}(K,H^{\ast }(\_))$. Along the way, we prove that this algebra of primitive elements is also noetherian.