AI 中文总结
研究从核模对称幺半范畴重构形式概型的方法，通过恢复挠子范畴并结合巴尔默谱来重构，还证明了特定情形下相关逆变函子的完全忠实性。
AI 中文摘要
我们为形式概型提供了一种部分函子性且具构造性的从核模对称幺半范畴进行重构的程序。具体而言，对于形式概型\(\mathfrak{X}\)，我们表明挠子范畴\(D_{\mathrm{tors}}(\mathfrak{X})\)可作为埃菲莫夫范畴\(\mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})\)的极大强紧生成局部化张量理想被恢复，克劳森 - 朔尔策范畴\(\mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})\)同理。结合巴尔默谱，我们从相应的核模对称幺半范畴重构\(\mathfrak{X}\)。此外，对于在域或\(\mathbb{Z}\)上拓扑有限型的形式概型，逆变函子\(\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})\)是完全忠实的；在仿射情形下，\(\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})\)有类似结论。
英文摘要
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 $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$.
