Blow-ups and normal bundles in connective and nonconnective derived geometries

Abstract

This work presents a generalization of derived blow-ups and of the derived deformation to the normal bundle from derived algebraic geometry to any geometric context. The latter is our proposed globalization of a derived algebraic context, itself a generalization of the theory of simplicial commutative rings and due to Bhatt–Mathew and Raksit.

One key difference between a geometric context and ordinary derived algebraic geometry is that the coordinate ring of an affine object in the former is not necessarily connective. When constructing generalized blow-ups, this not only turns out to be remarkably convenient, but also leads to a wider existence result. Indeed, we show that the derived Rees algebra and the derived blow-up exist for any affine morphism of stacks in a given geometric context. However, in general the derived Rees algebra will no longer be connective, hence in general the derived blow-up will not live in the connective part of the theory. Unsurprisingly, this can be solved by restricting the input to closed immersions. The proof of the latter statement uses a derived deformation to the normal bundle in any given geometric context, which is also of independent interest.

Besides the geometric context which extends algebraic geometry, the second main example of a geometric context is an extension of analytic geometry as based on categories of Ind-Banach spaces or modules. The latter is a recent construction, and includes many different flavors of analytic geometry, such as complex analytic geometry, non-Archimedean rigid analytic geometry and analytic geometry over the integers. The present work thus provides derived blow-ups and a derived deformation to the normal bundle in all of these, which is expected to have many applications.