Affineness and reconstruction in higher Zariski geometry

2025/10/19 by Anish Chedalavada, Chedalavada, Anish
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2510.16964

openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

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 Spec 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" V_G such that Perf_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.

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Related

• Higher Zariski Geometry
• Affineness and reconstruction in complex-periodic geometry
• A representation for algebraic K-theory of quasi-coherent modules over affine spectral schemes
• A Dual Representation in Spectral Algebraic Geometry
• Quasi-compact group schemes, Hopf sheaves, and their representations