Localizing invariants of inverse limits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929705707372544 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | In this paper we study the category of nuclear modules on an affine formal scheme as defined by Clausen and Scholze \cite{CS20}. We also study related constructions in the framework of dualizable and rigid monoidal categories. We prove that the $K$-theory (in the sense of \cite{E24}) of the category of nuclear modules on $\operatorname{Spf}(R^{\wedge}_I)$ is isomorphic to the classical continuous $K$-theory, which in the noetherian case is given by the limit $\varprojlim\limits_{n} K(R/I^n).$ This isomorphism was conjectured previously by Clausen and Scholze.
More precisely, we study two versions of the category of nuclear modules: the original one defined in \cite{CS20} and a different version, which contains the original one as a full subcategory. For our category $\operatorname{Nuc}(R^{\wedge}_I)$ we give three equivalent definitions. The first definition is by taking the internal $\operatorname{Hom}$ in the category $\operatorname{Cat}_R^{\operatorname{dual}}$ of $R$-linear dualizable categories. The second definition is by taking the rigidification of the usual $I$-complete derived category of $R.$ The third definition is by taking an inverse limit in $\operatorname{Cat}_R^{\operatorname{dual}}.$ For each of the three approaches we prove that the corresponding construction is well-behaved in a certain sense.
Moreover, we prove that the two versions of the category of nuclear modules have the same $K$-theory, and in fact the same finitary localizing invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localizing invariants of inverse limits Efimov, Alexander I. K-Theory and Homology Algebraic Geometry Algebraic Topology Category Theory Number Theory 19D55 In this paper we study the category of nuclear modules on an affine formal scheme as defined by Clausen and Scholze \cite{CS20}. We also study related constructions in the framework of dualizable and rigid monoidal categories. We prove that the $K$-theory (in the sense of \cite{E24}) of the category of nuclear modules on $\operatorname{Spf}(R^{\wedge}_I)$ is isomorphic to the classical continuous $K$-theory, which in the noetherian case is given by the limit $\varprojlim\limits_{n} K(R/I^n).$ This isomorphism was conjectured previously by Clausen and Scholze. More precisely, we study two versions of the category of nuclear modules: the original one defined in \cite{CS20} and a different version, which contains the original one as a full subcategory. For our category $\operatorname{Nuc}(R^{\wedge}_I)$ we give three equivalent definitions. The first definition is by taking the internal $\operatorname{Hom}$ in the category $\operatorname{Cat}_R^{\operatorname{dual}}$ of $R$-linear dualizable categories. The second definition is by taking the rigidification of the usual $I$-complete derived category of $R.$ The third definition is by taking an inverse limit in $\operatorname{Cat}_R^{\operatorname{dual}}.$ For each of the three approaches we prove that the corresponding construction is well-behaved in a certain sense. Moreover, we prove that the two versions of the category of nuclear modules have the same $K$-theory, and in fact the same finitary localizing invariants. |
| title | Localizing invariants of inverse limits |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology Category Theory Number Theory 19D55 |
| url | https://arxiv.org/abs/2502.04123 |