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.

我们从Henn-Lannes-Schwartz的意义上研究连通Noetherian不稳定代数R的拓扑无能度。当R是一个紧Lie群的mod p同调环时，Kuhn展示了如何通过基本阿贝尔p-子群的扶正器控制这个不变性。通过替换基本阿贝尔的扶正p与拉纳的部件-subgroups Ť-functor，并利用Steenrod代数上的不稳定代数技术，我们可以将Kuhn的结果推广到一大类相连的Noetherian不稳定代数。我们展示了如何将Kuhn的结果推广到更通用的组类别，例如有限虚拟同调维数组，有限组和Kac-Moody组。实际上，我们的结果适用范围更广，例如，对于具有Noetherian mod p的连通H空间，我们建立了Broto–Levi–Oliver意义上的p-局部紧致群的结果作用于流形上的紧凑李群的Borel等变同调。一路上，我们建立了一些独立利益的结果。例如，在最小深度的Noetherian不稳定代数的情况下，我们公式化并证明了Carlson深度猜想的一种形式。