Localizing invariants of inverse limits
Alexander I. Efimov
TL;DR
The paper develops a robust categorified approach to continuous K-theory for formal schemes by analyzing two dualizable nuclear-module categories, Nuc^{CS}(R^∧_I) and Nuc(R^∧_I). It provides three equivalent descriptions of Nuc(R^∧_I) via dualizable internal Hom, rigidification of the I-complete derived category, and inverses limits in dualizable Cat, and proves that their continuous K-theories coincide with lim_n K(R/I^n). A central technical achievement is the internal projectivity of certain I-torsion/ I-complete categories, which allows the computation of localizing invariants under inverse limits, including strong Mittag-Leffler systems. The results generalize classical continuous K-theory to non-noetherian settings through Koszul DG-algebras and prove that rigidification and nuclear objects yield compatible invariants, with applications to Hochschild homology and IndCoh-type contexts. Overall, the work provides a canonical, functorial categorification of continuous K-theory for formal schemes and clarifies when various nuclear-categorical constructions yield the same invariants and limits.
Abstract
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.