Localising invariants in derived bornological geometry
Jack Kelly, Devarshi Mukherjee
TL;DR
The paper develops analytic stacks in a bornological framework by extending the Toën–Vezzosi approach to derived geometry to complete bornological modules over Banach rings, and proves Nisnevich descent for bornological K-theory across Archimedean and non-Archimedean settings. It establishes a robust, largely topological and homotopical foundation via relative pre-geometries, descendable topologies, and geometric stacks, then specializes to bornological contexts to model derived rigid analytic spaces, overconvergent (dagger) analytic spaces, and complex analytic geometry. It introduces and analyzes derived algebras in the bornological setting, showing that quasi-coherent descent holds under étale and rational topologies, and proves a Grothendieck–Riemann–Roch-type statement for derived dagger analytic spaces. The work unifies complex and non-Archimedean analytic geometries within a single, flexible framework and provides comparisons to Porta–Yue Yu’s derived complex analytic geometry, thereby enabling formal geometric methods (e.g., Raynaud-style formal geometry) to be applied in bornological analytic contexts with broad potential for further applications in non-Archimedean and complex analytic geometry.
Abstract
We study several categories of analytic stacks relative to the category of bornological modules over a Banach ring. When the underlying Banach ring is a non-Archimedean valued field, this category contains derived rigid analytic spaces as a full subcategory. When the underlying field is the complex numbers, it contains the category of derived complex analytic spaces. In the second part of the paper, we consider localising invariants of rigid categories associated to bornological algebras. The main results in this part include Nisnevich descent for derived analytic spaces and a version of the Grothendieck-Riemann-Roch Theorem for derived dagger analytic spaces over an arbitrary Banach ring.