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Affineness and reconstruction in higher Zariski geometry

Anish Chedalavada

TL;DR

The work develops a higher Zariski geometry framework using 2-rings to study qcqs spectral and Dirac spectral schemes. It proves an affineness criterion for rigid 2-schemes and establishes a universal property identifying Perf_X as the affine global sections of the relative spectrum Spec^2_1 X, yielding a canonical reconstruction map gamma_X that recovers X from Perf_X in the qcqs case. This enables a geometrization of 2-rings, embedding 2-rings into spectra of affine 2-schemes and applying the construction to represent and compare tensor-triangulated phenomena, including spectral support varieties and stable module categories. The approach unifies Balmer–Thomason reconstruction with modern derived and spectral-algebraic methods and extends naturally to Dirac spectral contexts, broadening the range of moduli problems that admit affine 2-scheme descriptions. Overall, the paper provides a coherent toolkit for translating between tensor-triangular data and geometric objects in a higher-algebraic setting with concrete applications to representation theory and homotopy-theoretic contexts.

Abstract

We explain how the geometric framework introduced in arXiv:2508.11621 [math.AG] provides a universal property for the 2-rings of perfect complexes on qcqs spectral or Dirac spectral schemes. As an application, given a qcqs spectral or Dirac spectral scheme $X$ this produces a comparison morphism from $\operatorname{Spec} \mathrm{Perf}_{X}$ to $X$ itself, which is moreover natural in $X$. When $X$ is an ordinary qcqs scheme, this construction supplies a new proof of the Balmer-Thomason reconstruction of $X$ from its space of thick subcategories, assuming the result for noetherian rings due to Neeman. As another application, we find spectral and Dirac spectral enhancements of support varieties arising for 2-rings in representation theory which "geometrize" the 2-rings that produce them. For example, given a finite group $G$ over a field $k$, this produces a "spectral support variety" $\mathcal{V}_{G}$ such that $\mathrm{Perf}_{\mathcal{V}_{G}}$ maps into the stable module category of $kG$. We derive these results as a corollary of a general affineness criterion for 2-schemes which are covered by the Zariski spectra of rigid 2-rings: this states that such 2-schemes are affine if and only if they are quasicompact and quasiseparated.

Affineness and reconstruction in higher Zariski geometry

TL;DR

The work develops a higher Zariski geometry framework using 2-rings to study qcqs spectral and Dirac spectral schemes. It proves an affineness criterion for rigid 2-schemes and establishes a universal property identifying Perf_X as the affine global sections of the relative spectrum Spec^2_1 X, yielding a canonical reconstruction map gamma_X that recovers X from Perf_X in the qcqs case. This enables a geometrization of 2-rings, embedding 2-rings into spectra of affine 2-schemes and applying the construction to represent and compare tensor-triangulated phenomena, including spectral support varieties and stable module categories. The approach unifies Balmer–Thomason reconstruction with modern derived and spectral-algebraic methods and extends naturally to Dirac spectral contexts, broadening the range of moduli problems that admit affine 2-scheme descriptions. Overall, the paper provides a coherent toolkit for translating between tensor-triangular data and geometric objects in a higher-algebraic setting with concrete applications to representation theory and homotopy-theoretic contexts.

Abstract

We explain how the geometric framework introduced in arXiv:2508.11621 [math.AG] provides a universal property for the 2-rings of perfect complexes on qcqs spectral or Dirac spectral schemes. As an application, given a qcqs spectral or Dirac spectral scheme this produces a comparison morphism from to itself, which is moreover natural in . When is an ordinary qcqs scheme, this construction supplies a new proof of the Balmer-Thomason reconstruction of from its space of thick subcategories, assuming the result for noetherian rings due to Neeman. As another application, we find spectral and Dirac spectral enhancements of support varieties arising for 2-rings in representation theory which "geometrize" the 2-rings that produce them. For example, given a finite group over a field , this produces a "spectral support variety" such that maps into the stable module category of . We derive these results as a corollary of a general affineness criterion for 2-schemes which are covered by the Zariski spectra of rigid 2-rings: this states that such 2-schemes are affine if and only if they are quasicompact and quasiseparated.
Paper Structure (22 sections, 28 theorems, 79 equations, 1 figure)

This paper contains 22 sections, 28 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Schemes in classical algebraic geometry
  3. Affine moduli in higher Zariski geometry
  4. Reconstruction of schemes and geometrization of 2-rings
  5. Overview
  6. Conventions
  7. Acknowledgements
  8. Recollections on the Geometry of Rings and 2-Rings
  9. Geometries
  10. 2-rings
  11. Prerequisite results on the Zariski geometry of 2-rings
  12. Preliminaries on relative spectra and descent
  13. Étale maps
  14. Localic structured topoi
  15. Relative spectra of spectral schemes are 0-localic
  16. ...and 7 more sections

Key Result

Theorem 1

Let $(X, \mathcal{O}_{X})$ be a rigid $2$-scheme. Then $(X, \mathcal{O}_{X})$ is an affine $2$-scheme if and only if the underlying topological space $X$ is quasicompact and quasiseparated (or qcqs).

Figures (1)

  • Figure 1: Creation of the Birds, Remedios Varo, 1957

Theorems & Definitions (97)

  • Example : Complex projective space
  • Example : Projective space
  • Definition
  • Theorem 1
  • Definition
  • Theorem 2
  • Remark 1
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • ...and 87 more