高阶代数$K$ -群与$\mathcal D$ -分裂序列

Comparing motives of smooth algebraic varieties

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq 0,$$ are quasi-isomorphisms, it is proved that for every $k$-smooth algebraic variety $X$ the morphisms of twisted motives of $X$ with $\mathbb Z[1/e]$-coefficients $$M_{\mathcal A}(X)(q)\otimes\mathbb Z[1/e]\to M_{\mathcal B}(X)(q)\otimes\mathbb Z[…

Algebraic K-theory, K-regularity, and T-duality of $\mathcal{O}_\infty$-stable $C^*$-algebras

We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an $E_\infty$-operad starting from any strongly self-absorbing $C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\bf K}_Σ^{top}(-…

代数无理旋转代数与$SL(2,\mathbb Z)$有限子群交叉积的Hochschild和循环同调

Hochschild and cyclic homology of the crossed product of algebraic irrational rotational algebra by finite subgraoups of $SL(2,\mathbb Z)$

设$Γ\subset SL(2, \mathbb Z)$是作用在无理旋转代数$\mathcal A_θ$上的有限子群，通过对$SL(2,\mathbb Z)$的正则作用的限制.考虑由代数无理旋转代数上的$Γ$作用的限制得到的交叉积代数$\mathcal A_θ^{alg} \rtimes Γ$.本文证明了交叉积代数$\mathcal A_θ^{alg} \rtimes Γ$的同调群的若干结果。我们还分析了光滑交叉积代数$\mathcal A_θ\rtimes Γ$的情形，并计算了它的一些同调群。

Twisted K-theory, Real $\mathcal{A}$-bundles and Grothendieck-Witt groups

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted $K$-theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties.

射影复曲面格式的代数K理论的一个分裂结果

A splitting result for the algebraic K-theory of projective toric schemes

设X是定义在交换环R上的投射环面概型，R上有一个充线丛L.我们证明了它的K-理论有k+1个直和项K（R），其中k是非负整数中最小的，使得扭曲线丛L（-k-1）不是非循环的.事实上，使用组合描述的拟相干层，我们证明了结果的环R是交换的，否则左诺特。

代数 K-理论单位的谱

The spectrum of units of algebraic $K$-theory

本文研究代数 K-理论单位谱的 Postnikov 截断分裂，构造仿射概形上 [0,1]-截断的层分裂，并计算 K(Z)、K(F_ℓ) 的严格单位连通谱，同时证明其作为预层乃至逐点均不分裂。