AI 中文总结
该研究证明了刚性解析空间的连续K-理论与解析K-理论在$\boldsymbol{\rm A}^1$-同伦范畴中可表示，还得到轻凝聚谱系数的可表示性结论，并证明了Weibel消失定理及连续K-理论在局部Tate对上的$\boldsymbol{\rm A}^1$-不变性。
AI 中文摘要
我们证明刚性解析空间的连续K-理论与解析K-理论（Kerz--Saito--Tamme型）均满足关于Nisnevich拓扑的下降性。结合假设奇点解消时它是$\boldsymbol{\rm A}^1$-不变的这一事实，我们推导出它在刚性空间的$\boldsymbol{\rm A}^1$-同伦范畴（Dahlhausen--Yaylali型）中是可表示的。我们将该表示对象与$\boldsymbol{\rm Z}\times{\rm BGL}$以及代数K-理论的解析化等同起来。由此，我们得到了轻凝聚谱系数的可表示性结论。此外，我们证明了Weibel消失定理，且连续K-理论在局部Tate对上是$\boldsymbol{\rm A}^1$-不变的（无任何正则性假设）。
英文摘要
We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (à la Kerz--Saito--Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is $\mathbb{A}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the $\mathbb{A}^{1}$-homotopy category of rigid spaces (à la Dahlhausen--Yaylali). We identifiy the representing object with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is $\mathbb{A}^1$-invariant on local Tate pairs (without any regularity assumption).
Comments52 pages. Split-off from arXiv:2407.09606v2 with new coauthor, corrected proofs, and new results. Comments very welcome!
