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Algebraic Models for Quasi-Coherent Sheaves in Spectral Algebraic Geometry

Adam Pratt

TL;DR

The work develops an algebraic model framework for the category of quasi-coherent sheaves on non-connective geometric spectral stacks by leveraging adapted homology theories and Franke's algebraicity theorem. It identifies $\mathrm{QCoh}(\mathfrak X)$ with comodules over a coalgebra $\Gamma$ and uses the adapted homology theory $\pi_*$ on $\mathrm{Comod}_{\Gamma}$ to produce equivalences with the periodic derived category on comodules, $D^{\mathrm{per}}(\mathrm{Comod}_{\Gamma_*})$, in a controlled truncation. The paper provides explicit conditions under which QCoh on stacks like flat projective spaces and chromatic loci admit algebraic models, and it offers a new algebro-geometric proof of chromatic algebraicity via $E(n)_*E(n)$-comodules and moduli stacks of oriented formal groups. This bridges spectral algebraic geometry and derived algebraic geometry, enabling algebraic techniques to study stable homotopy categories through concrete abelian-model frameworks.

Abstract

In this paper we prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology theories.

Algebraic Models for Quasi-Coherent Sheaves in Spectral Algebraic Geometry

TL;DR

The work develops an algebraic model framework for the category of quasi-coherent sheaves on non-connective geometric spectral stacks by leveraging adapted homology theories and Franke's algebraicity theorem. It identifies with comodules over a coalgebra and uses the adapted homology theory on to produce equivalences with the periodic derived category on comodules, , in a controlled truncation. The paper provides explicit conditions under which QCoh on stacks like flat projective spaces and chromatic loci admit algebraic models, and it offers a new algebro-geometric proof of chromatic algebraicity via -comodules and moduli stacks of oriented formal groups. This bridges spectral algebraic geometry and derived algebraic geometry, enabling algebraic techniques to study stable homotopy categories through concrete abelian-model frameworks.

Abstract

In this paper we prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology theories.
Paper Structure (13 sections, 17 theorems, 56 equations)

This paper contains 13 sections, 17 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Acknowlegements
  4. Background
  5. Spectral algebraic geometry
  6. Adapted homology theories
  7. The algebraicity theorem
  8. Algebraic models in spectral algebraic geometry
  9. Adapted homology theories and nonconnective geometric spectral stacks
  10. The algebraic model
  11. Examples
  12. Projective space
  13. Chromatic algebraicity

Key Result

Theorem 23

Suppose that $H:\mathcal{C} \to \mathcal{A}$ is a conservative, adapted homology theory, and assume that Then, we have an equivalence of homotopy $(q+1-d)$-categories

Theorems & Definitions (59)

  • Remark 1
  • Definition 2: Faithful flatness, Ma18
  • Remark 3
  • Definition 4
  • Remark 5
  • Definition 6: Quasi-coherent sheaves
  • Example 7
  • Remark 8
  • Definition 9
  • Definition 10
  • ...and 49 more