Reconstruction of Formal Schemes from Categories of Nuclear Modules

Abstract

We provide a partially functorial and constructive reconstruction procedure for formal schemes from symmetric monoidal categories of nuclear modules. More precisely, for a formal scheme $\mathfrak{X}$, we show that the torsion subcategory $D_{\mathrm{tors}}(\mathfrak{X})$ can be recovered as the maximal strongly compactly generated localizing tensor ideal of Efimov's category $\mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$, and similarly for the Clausen--Scholze category $\mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$. Combining this with the Balmer spectrum, we reconstruct $\mathfrak{X}$ from the corresponding symmetric monoidal category of nuclear modules. Moreover, for formal schemes topologically of finite type over a field or over $\mathbb{Z}$, the contravariant functor $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$ is fully faithful; in the affine case, the analogous statement holds for $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$.