Localizing invariants of inverse limits

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.