半赋值环的正则性与代数K-理论的同伦不变性

Henselian 赋值环的 Grothendieck--Witt 群

Grothendieck--Witt Groups of Henselian Valuation Rings

本文证明代数 K-理论及高阶 Grothendieck--Witt 群在 Henselian 赋值环上满足局部常值条件，即有限系数下典范剩余映射为同构，适用于等特征与混合特征，证明基于 Suslin 方法的修改。

The K-theory of perfectoid rings

We establish various properties of the p-adic algebraic K-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the p-adic K-theory of such rings is homotopy invariant, and coincides with the p-adic K-theory of the p-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł.

非阿基米德双变K-理论

Nonarchimedean bivariant K-theory

本文为非阿基米德bornological代数构建了双变K-理论，作为dagger同伦不变、矩阵稳定和切除函子的万有目标，并由此得到满足dagger同伦不变性、矩阵稳定性和切除性的稳定化过收敛解析K-理论。

Algebraic K-theory of the infinite place

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also show that the "residue field at infinity" is badly behaved from a K-theoretic point of view.

K-theory of non-archimedean rings I

We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms models. Along the way we also prove some results about the algebraic K-theory of Tate rings.

Towards Vorst's conjecture in positive characteristic

Vorst's conjecture relates the regularity of a ring with the $\mathbb{A}^1$-homotopy invariance of its $K$-theory. We show a variant of this conjecture in positive characteristic.