$\mathbf Z_2$-graded Cech Cohomology in Noncommutative Geometry

$\mathrm{RO}(\mathbb T)$-graded $\mathrm{TF}$ of perfectoid rings

For a perfectoid ring $R$, we compute the full $\mathrm{RO}(\mathbb T)$-graded ring $\mathrm{TF}_\bigstar(R;\mathbf Z_p)$. This extends and simplifies work of Gerhardt and Angeltveit-Gerhardt. In even degrees, we find an $\mathrm{RU}(\mathbb T)$-graded version of Bökstedt periodicity, with some additional classes in the case of perfect $\mathbf F_p$-algebras. In odd degrees, we find extremely intricate and rather mysterious torsion. We also discuss the $\mathrm{RO}(\mathbb T)$-graded homotopy Tambara functors $\underlineπ_\bigstar\mathrm{THH}(R;\mathbf Z_p)$.

Twistings of KR for Real groupoids

B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some equivariant Cech cohomology theory. The construction of the twisted KR-functor is outlined.

分次赋值除环的群$\mathrm{TK}_1$

The group $\mathrm{TK}_1$ of graded and valued division algebras

研究分次赋值除环的挠群$\mathrm{TK}_1$，给出描述$\mathrm{TK}_1(E)$的正合序列及显式公式，确定阻碍群$\mathbf{H}$，证明短正合序列和稳定性定理，还得到挠怀特黑德群相关结果的分次类似物。

Box operads与更高Gerstenhaber括号

Box operads and higher Gerstenhaber brackets

引入描述矩形标记盒子演算的对称operad $\square p$，证明其作用于quiver的Gerstenhaber-Schack对象并诱导出控制lax prestack形变理论的 $L_{\infty}$-结构。

关于无 $\mathbb A^1$-不变性的动机球谱的注记

Remarks on the motivic sphere without $\mathbb A^1$-invariance

本文将动机球谱的基本性质推广到非 $\mathbb A^1$-不变的动机谱范畴，证明 Milnor-Witt K-理论关系成立，并给出有理动机上同调谱 $\mathrm H\mathbb Q$-模的多种等价刻画。