论文标题
Noetherian不稳定代数的拓扑结构
The topological nilpotence degree of a Noetherian unstable algebra
论文作者
论文摘要
我们研究了一个连接的Noetherian不稳定代数$ r $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ R $的拓扑nilpotence学位。当$ r $是紧凑型谎言组的mod $ p $共同体学戒指时，库恩展示了这种不变的如何受到基础abelian $ p $ subgroups的中央化的控制。通过更换具有Lannes的$ t $ functor组件的小学Abelian $ P $ - 群体的中心化，并利用steenrod代数上不稳定的代数的技术，我们可以将Kuhn的结果推广到Kuhn的一大批连接的Notethian notethian不稳定稳定性的Elgebras。我们展示了这是如何将Kuhn的结果推广到更多一般类别的群体的结果，例如有限的虚拟共同体学维度，Profinite组和KAC-MOODY群体。实际上，我们的结果更广泛地适用，例如，我们从Broto-Levi-Oliver的意义上为$ p $ local compact群体建立了结果，用于与Noetherian mod $ p $同居的连接的$ h $ spaces，以及Borel Equivariant同胞在薄compact compot to compact of compart compact of compact of compare。在此过程中，我们建立了几个独立关注的结果。例如，在赛义利亚的不稳定代数的最小深度代数的情况下，我们制定并证明了卡尔森深度猜想的一种版本。
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.
