Approximation of the centre of unstable algebras using the nilpotent filtration

Abstract

In a precedent article, we computed the set $\textbf{C}(K)$ of central elements of an unstable algebra $K$ over the Steenrod algebra, in the sense of Dwyer and Wilkerson, when $K$ is noetherian and $nil_1$-closed. For $K$ noetherian and $k$ a positive integer, we define $\textbf{C}_k(K)$, the set of so-called central elements of $K$ away from $\mathcal{N}il_k$ in such a way that, for $K$ $nil_k$-closed, $\textbf{C}(K)=\textbf{C}_k(K)$. The sets $\textbf{C}_k(K)$ are a decreasing filtration, and we describe the obstruction for an element in $\textbf{C}_k(K)$ to be in $\textbf{C}_{k+1}(K)$. Since, for $K$ noetherian, $K$ is always $nil_k$-closed for $k$ big enough, this gives us a way to compute the set of central elements of $K$.