In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space $\mathbf{R}\text{Hilb}(X)$ for a separated $k$-analytic space $X$. Such representability results rely on a localization theorem stating that if $\mathfrak{X}$ is a quasi-compact and quasi-separated formal scheme, then the $\infty$-category $\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$ of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the $\infty$-category $\text{Coh}^{-}(\mathfrak{X})$. Along the way, we prove several results concerning the $\infty$-categories of formal models for almost perfect modules on derived $k$-analytic spaces.

中文翻译：

---

派生的非阿基米德分析希尔伯特空间

在这篇简短的论文中，我们结合了 [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017,arXiv:1704.01683; Porta 和 Yu，刚性解析几何中的衍生 Hom 空间，预印本，2018，arXiv:1801.07730] 与 [António,$p$-adic 导出的形式几何和导出的雷诺定位定理，预印本，2018，arXiv:1805.03302] 来证明导出的希尔伯特空间的存在性可表示性$\mathbf{R}\text{Hilb}(X)$对于一个分开的$k$-分析空间$X$. 这种可表示性结果依赖于一个定位定理，该定理表明，如果$\mathfrak{X}$是一个准紧准分离形式方案，则$\infty$-类别$\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$在通用纤维上几乎完美的配合物可以实现为的 Verdier 商$\infty$-类别$\text{Coh}^{-}(\mathfrak{X})$. 在此过程中，我们证明了关于$\infty$-派生上几乎完美模块的形式模型类别$k$-分析空间。