Localising invariants in derived bornological geometry

TL;DR

This paper develops a framework for derived analytic spaces over Banach rings, unifying rigid and complex analytic geometries, and establishes descent and Riemann-Roch theorems for these spaces.

Contribution

It introduces categories of derived analytic spaces relative to bornological modules and proves descent and Riemann-Roch results in this new setting.

Findings

Nisnevich descent for derived analytic spaces

Grothendieck-Riemann-Roch theorem for derived dagger analytic spaces

Unification of rigid and complex analytic geometries

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.