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This work presents a generalization of derived blow-ups and of the deriveddeformation to the normal bundle from derived algebraic geometry to anygeometric context. The latter is our proposed globalization of a derivedalgebraic context, itself a generalization of the theory of simplicialcommutative rings.  One key difference between a geometric context and ordinary derived algebraicgeometry is that the coordinate ring of an affine object in the former is notnecessarily connective. When constructing generalized blow-ups, this not onlyturns out to be remarkably convenient, but also leads to a wider existenceresult. Indeed, we show that the derived Rees algebra and the derived blow-upexist for any affine morphism of stacks in a given geometric context. However,in general the derived Rees algebra will no longer be connective, hence ingeneral the derived blow-up will not live in the connective part of the theory.Unsurprisingly, this can be solved by restricting the input to closedimmersions. The proof of the latter statement uses a derived deformation to thenormal bundle in any given geometric context, which is also of independentinterest.  Besides the geometric context which extends algebraic geometry, the secondmain example of a geometric context will be an extension of analytic geometry.The latter is a recent construction, and includes many different flavors ofanalytic geometry, such as complex analytic geometry, non-archimedean rigidanalytic geometry and analytic geometry over the integers. The present workthus provides derived blow-ups and a derived deformation to the normal bundlein all of these, which is expected to have many applications.

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