Bocksteins and the nilpotent filtration on the cohomology of spaces

Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here. An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules l…

Filtered Topological Cyclic Homology and relative K-theory of nilpotent ideals

In this paper we examine certain filtrations of topological Hochschild homology and topological cyclic homology. As an example we show how the filtration with respect to a nilpotent ideal gives rise to an analog of a theorem of Goodwillie saying that rationally relative K-theory and relative cyclic homology agree. Our variation says that the p-torsion parts agree in a range of degrees. We use it to compute K_i(Z/p^m) for i < p-2.

使用幂零滤逼近不稳定代数的中心

Approximation of the centre of unstable algebras using the nilpotent filtration

本文通过定义远离$\mathcal{N}il_k$的中心元素集合$\textbf{C}_k(K)$并利用其递减滤过结构，提出了一种计算noetherian不稳定代数K中心元素的方法。

A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations…

The nilpotent filtration and the action of automorphisms on the cohomology of finite p--groups

We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of th…

Infinite loop spaces and nilpotent K-theory

Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL_{\infty}(R)^{+}$ and $Q_0(\mathbb{S}^{0})$. We show that these infinite loop spaces are the zero spaces of non-unital $E_\infty$-ring spectra. We introduce the notion of $q$-nilpotent K-theory of a CW-complex $X$ for any $q\ge 2$, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by $\mathbb Z\times B(q,U)$, were $B(q,U)$ is the $q$-th t…