Some formal gluing diagrams for continuous K-theory

TL;DR

This paper develops a framework for constructing diagrams of stable categories that preserve continuous K-theory under certain fiber-cofiber sequences, enabling new insights into formal gluing and adelic descent in algebraic K-theory.

Contribution

It introduces a method to analyze diagrams of dualizable stable categories related to fiber-cofiber sequences, applying it to formal gluing and adelic descent in continuous K-theory.

Findings

Recovered Clausen--Scholze's gluing of continuous K-theory along punctured neighborhoods.

Verified a continuous adelic descent statement for localizing invariants.

Established a diagrammatic approach for dualizable categories in algebraic K-theory.

Abstract

We study a construction of diagrams of dualizable presentable stable $\infty$-categories associated with certain fiber-cofiber sequences over rigid bases, which are sent by localizing invariants, in particular continuous K-theory, to limit diagrams. We apply this to investigate two closely related types of diagrams pertinent to the formal gluing situation; we recover Clausen--Scholze's gluing of continuous K-theory along punctured tubular neighborhoods via Efimov's nuclear module category, and we verify a continuous version of adelic descent statement for localizing invariants on dualizable categories.