A note on ideals in derived geometries

TL;DR

This paper develops the theory of derived quasi-coherent ideals in stacks, compares notions of adic completeness, and explores applications like derived scheme-theoretic images and deformation spaces in derived algebraic geometry.

Contribution

It introduces a comprehensive framework for derived ideals in stacks, linking them to formal spectra, Rees algebras, and deformation theory, advancing the understanding of derived geometric structures.

Findings

Derived ideals relate to formal spectra and Rees algebras.

Deformation spaces of derived stacks are nonconnectively affine.

Constructed derived scheme-theoretic images in general settings.

Abstract

We develop the basic theory of derived quasi-coherent ideals for stacks relative to a given derived algebraic context. We compare different notions of adic completeness with respect to derived ideals, define and compare formal spectra and formal completions along closed immersions, and connect the theory of derived ideals to that of derived extended Rees algebras. A first application is the construction of derived scheme-theoretic images in full generality. We further show that the deformation space of any nonconnectively affine morphism of derived stacks is nonconnectively affine over the base. We close with a first exploration of transmutation cohomology and filtrations thereof in this more general context.