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A note on ideals in derived geometries

Zachary Gardner, Jeroen Hekking

TL;DR

This work develops a comprehensive framework for derived ideals in geometry relative to a fixed derived algebraic context, unifying completions, formal spectra, and Rees-algebra filtrations within a global stack-theoretic setting. By clarifying the correspondence between derived ideal pairs and arrows of derived algebras, the authors construct derived scheme-theoretic images and establish that deformation spaces of nonconnectively affine morphisms remain affine over the base, via extended Rees algebras. They formulate and compare multiple notions of completeness (I-complete vs I-adically complete) and prove their equivalence in the locally finitely generated case, linking formal spectra with formal completions. The paper then extends these ideas to derived $ ext{C}$-algebra stacks, introducing transmutation, Weil restrictions, and transmutation cohomology, which provide powerful tools for handling filtered/graded objects and their global and cohomological aspects in derived settings. Overall, the work provides a robust, global framework for derived ideals, filtrations, and deformations, enabling systematic construction of scheme-theoretic images and formal geometry in derived contexts with broad potential applications in prismatic cohomology and beyond.

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.

A note on ideals in derived geometries

TL;DR

This work develops a comprehensive framework for derived ideals in geometry relative to a fixed derived algebraic context, unifying completions, formal spectra, and Rees-algebra filtrations within a global stack-theoretic setting. By clarifying the correspondence between derived ideal pairs and arrows of derived algebras, the authors construct derived scheme-theoretic images and establish that deformation spaces of nonconnectively affine morphisms remain affine over the base, via extended Rees algebras. They formulate and compare multiple notions of completeness (I-complete vs I-adically complete) and prove their equivalence in the locally finitely generated case, linking formal spectra with formal completions. The paper then extends these ideas to derived -algebra stacks, introducing transmutation, Weil restrictions, and transmutation cohomology, which provide powerful tools for handling filtered/graded objects and their global and cohomological aspects in derived settings. Overall, the work provides a robust, global framework for derived ideals, filtrations, and deformations, enabling systematic construction of scheme-theoretic images and formal geometry in derived contexts with broad potential applications in prismatic cohomology and beyond.

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.

Paper Structure

This paper contains 34 sections, 59 theorems, 191 equations.

Table of Contents

  1. Background
  2. Derived algebraic contexts
  3. Graded objects
  4. Geometry over a given context
  5. Deformation to the normal bundle
  6. Virtual Cartier divisors
  7. Generalized filtered objects
  8. First definitions
  9. $\mathbb{M}$-filtered derived algebraic contexts
  10. The $\mathbb{N}$- and $\mathbb{Z}$-filtered case
  11. Ideal pairs: affine case
  12. Derived ideal pairs
  13. Rees algebras and $I$-adic filtrations
  14. Ideal pairs relative to a base
  15. Quasi-coherent ideals & Rees algebras
  16. ...and 19 more sections

Key Result

Theorem 2

Let $X$ be a $\mathscr{C}_{\geq 0}$-stack, $\mathcal{I}$ a quasi-coherent ideal on $X$, and $\mathcal{M}$ a quasi-coherent $\mathcal{O}_X$-module. If $\mathcal{M}$ is derived $\mathcal{I}$-adically complete then it is derived $\mathcal{I}$-complete (Corollary Cor:derived_complete_implies_adically).

Theorems & Definitions (168)

  • Theorem 2: Completeness
  • Theorem 3: Affines-as-ideals
  • Theorem 4: Formal spectrum is formal completion
  • Theorem 5: Affines are closed under deformation spaces
  • Theorem 6: Cotangent complex as derived conormal sheaf
  • Definition 1.1.1
  • Definition 1.4.2
  • Lemma 1.4.3
  • Definition 1.4.4
  • Proposition 1.4.5
  • ...and 158 more