Every continuous localizing invariant satisfies adelic descent when computed on nuclear modules over the adelic rings of a scheme of finite type over Z.
Some formal gluing diagrams for continuous K-theory
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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.
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Adelic descent for continuous localizing invariants
Every continuous localizing invariant satisfies adelic descent when computed on nuclear modules over the adelic rings of a scheme of finite type over Z.