分圆迹映射$K \rightarrow TC$与Grothendieck迹公式在非交换几何上的类比
Analogy between the cyclotomic trace map $K \rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry
摘要：
在这篇文章中，我们提出了一个分类程序，以捕捉之间的类比结晶Grothendieck-Lefschetz迹公式和分圆迹映射$K\rightarrow TC$从代数$K$ -理论的拓扑循环同调$TC$。首先，我们把这类方案归为非交换方案的$(2, \infty)$ -范畴。这给出了一个方案的有理点集的分类。然后，我们对Crystalline Grothendieck-Lefschetz迹公式进行了分类，并在$\mathbf{F}_{p}$上的非对易格式下找到了Crystalline上同调的一个类似形式.我们的类比表明，在非交换设置$l\neq p$的$l$ -进上同调迹公式的分类存在。最后，我们写下相应的字典。
英文摘要：
In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of noncommuative schemes over $\mathbf{F}_{p}$. Our analogy suggests the existence of a categorification of the $l$-adic cohomology trace formula in the noncommutative setting for $l\neq p$. Finally, we write down the corresponding dictionary.
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