The topological nilpotence degree of a Noetherian unstable algebra

Abstract

We investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

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

“…In [Kuh07] and [Kuh13], Kuhn used it to approximate the depth of K as well as invariants d 0 (K) and d 1 (K) introduced by Henn, Lannes and Schwartz in [HLS95], in the case where K is the cohomology of a group. Heard generalised those results for K noetherian in [Hea20] and [Hea21].…”

“…In [Hea21], Heard showed that for K noetherian, K admits a unique (up to isomorphism) central element (C, γ) such that γ induces a structure of finitely generated K-module on H * (C) and dim(C) is maximal among such central elements. Heard called this central element the centre of K. The centre of an unstable algebra have been shown to be an important invariant.…”

“…For K an unstable algebra, we recall the definition of the connected components of T V (K) which is exposed in [Hea20] and [Hea21]. Such a decomposition exists for any graded algebra over a p-boolean algebra.…”

“…The notion of a central element of an unstable algebra K is defined by Dwyer and Wilkerson in [DW92b] and they used it in [DW92a] to exhibit the only exotic finite loop space at the prime 2. The centre of K has been studied in details in [Hea20] and [Hea21]. The aim of this subsection is to recall some known facts about central elements of an unstable algebra.…”

“…We recall the definition of the centre of a noetherian unstable algebra from [Hea21] which is in some sense the maximal regular central element of K.…”