关于有限特征环的相对和双相对K理论

局部环上线性群的相对完备化与上同调

Relative completions and the cohomology of linear groups over local rings

本文研究了域$k$上约化代数群中相对于Zebraki稠密表示的群的完备化问题。特征零情况以前由R. Hain;我们将他的结果推广到任意特征。主要应用是研究群的上同调，如$SL_n(k[[T]])$。

The $K$-theory of toric schemes over regular rings of mixed characteristic

We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.

特征为2的八阶四元数群的Hochschild上同调环

On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two

设$k$是特征为2的代数闭域，$Q_8$是$8$阶四元数群.确定了群代数$kQ_8$的Hochschild上同调环上的Gerstenhaber李代数结构和Batalin-Vilkovisky结构.

A Bloch-Ogus Theorem for henselian local rings in mixed characteristic

We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $\mathbb{A}^1$-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application we derive a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology over a henselian discrete valuation ring with infinite residue field.

Witt vectors with coefficients and characteristic polynomials over non-commutative rings

For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that W(R) := W(R;R) is Morita invariant in R. For an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element $χ_f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The ass…

相对于δ环的棱柱上同调

Prismatic cohomology relative to $δ$-rings

本文发展了相对于δ环的棱柱和syntomic上同调，证明了相对棱柱上同调实际上不依赖棱柱结构，仅依赖于底层δ环。提供了三种等价的棱柱上同调定义，并应用于滤环的棱柱上同调理论。