AI 中文总结
该研究计算连通莫拉瓦K理论\(k(n)\)在特定度数下积分代数K理论群的基数，通过轨道过滤等方法，结合对特定形式DGA的分析及动机过滤，得出相关基数结果，还计算了\(W(\overline{\mathbb{F}}_p)/p^n\)代数K理论群基数，其偶数度数下消失。
AI 中文摘要
我们计算了连通莫拉瓦K理论\(k(n)\)在所有模\(2p - 2\)不与\(0\)或\(1\)同余的度数下的积分代数K理论群的基数。将\(k(n)\)的系数基变换到代数闭包\(\overline{\mathbb{F}}_p\)后，我们确定了所有度数下的相应基数，并表明这些群在偶数度数下消失。我们的方法使用了由拓扑霍赫希尔德同调的梅过滤产生的拓扑循环同调上的轨道过滤。结合对\(\mathbb{F}[x_{2m}]\)形式的形式DGA的拓扑循环同调的分析（其中\(\mathbb{F}\)是特征\(p\)的完美域），我们证明了具有同伦\(\mathbb{F}[x_{2m}]\)的\(\mathbb{E}_1\)环的拓扑循环同调的强基数结果。对于有限域上的莫拉瓦K理论，我们进一步使用了哈恩 - 拉克西特 - 威尔逊的动机过滤。作为我们方法的一个应用，我们计算了\(\overline{\mathbb{F}}_p\)的\(p\)典型维特向量的截断\(W(\overline{\mathbb{F}}_p)/p^n\)的代数K理论群的基数；特别地，它们在偶数度数下消失。
英文摘要
We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn--Raksit--Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.
