Approximation of the centre of unstable algebras using the nilpotent filtration

Abstract

In a precedent article, we computed the set $\text{C}(K)$ of central elements of an unstable algebra $K$ over the Steenrod algebra, in the sense of Dwyer and Wilkerson, when $K$ is noetherian and $ni{l}_1$-closed. For $K$ noetherian and $k$ a positive integer, we define ${\text{C}}_k(K)$, the set of so-called central elements of $K$ away from $\mathcal{N}i{l}_k$ in such a way that, for $K$ $ni{l}_k$-closed, $\text{C}(K)={\text{C}}_k(K)$. The sets ${\text{C}}_k(K)$ are a decreasing filtration, and we describe the obstruction for an element in ${\text{C}}_k(K)$ to be in ${\text{C}}_{k+1}(K)$. Since, for $K$ noetherian, $K$ is always $ni{l}_k$-closed for $k$ big enough, this gives us a way to compute the set of central elements of $K$.

29 pages