Ben-Bassat, Oren and Hekking, Jeroen (2025) Blow-ups and normal bundles in connective and nonconnective derived geometries. ADVANCES IN MATHEMATICS, 480 (Part C): 110530. ISSN 0001-8708, 1090-2082
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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. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Derived algebraic geometry; Homotopy algebraic geometry; Derived blow-ups |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 15 Apr 2026 05:46 |
| Last Modified: | 15 Apr 2026 05:46 |
| URI: | https://pred.uni-regensburg.de/id/eprint/66676 |
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