论文标题
通勤环光谱的拓扑循环同源性的动机过滤
A motivic filtration on the topological cyclic homology of commutative ring spectra
论文作者
论文摘要
对于Prime数字$ P $和A $ p $ - QuasisyNtomic的通勤环$ r $，bhatt-morrow- scholze-scholze定义的动机过滤，$ \ mathrm {thh}（thh}（thh}（r）的$ p $ -completions， $ \ mathrm {tc}（r）$，与$ \ mathrm {tp}（r）$和$ \ mathrm {tc}（r）$分别恢复了$ r $的棱镜和构想同胞的相关对象。我们提供了这些过滤的替代结构，当$ r $是行为良好的通勤环频谱时，我们也适用。例如，我们可以将$ r $带到$ \ mathbb {s} $，$ \ mathrm {mu} $，$ \ mathrm {ku} $，$ \ mathrm {ko} $，或$ \ mathrm {tmrm {tmf} $。我们计算Adams Summand $ \ ell $的Mod $（P，V_1）$ syntomic共同体，并观察到，当$ v（1）_*\ Mathrm {tc}（\ ell）$ collapses $ v（1）_**\ Mathrm {tc} $ v（1）_*\ mathrm {tc} $ collapses的动机频谱序列时
For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.
