Higher Zariski Geometry

Abstract: We revisit the classical constructions of tensor-triangular geometry in the setting of stably symmetric monoidal idempotent-complete $\infty$-categories, henceforth referred to as 2-rings. In this setting, we produce a Zariski topology, a Zariski spectrum, a category of locally 2-ringed spaces (more generally $\infty$-topoi), and an affine spectrum-global sections adjunction, based on the framework of $\infty$-topoi with geometric structure'' as developed by Lurie in \cite{LurieDAG5}. Using work of Kock and Pitsch, we compute that the underlying space of the Zariski spectrum of a 2-ring recovers the Balmer spectrum of its homotopy category. These constructions mirror the analogous structures in the classical Zariski geometry of commutative rings (and commutative ring spectra), and we also demonstrate additional compatibility between classical Zariski and higher Zariski geometry. For rigid 2-rings, we show that the descent results of Balmer and Favi admit coherent enhancements. As a corollary, we obtain that the Zariski spectrum fully faithfully embeds rigid 2-rings into locally 2-ringed $\infty$-topoi. In an appendix, we prove astalk-locality principle'' for the telescope conjecture in the rigid setting, extending earlier work of Hrbek.

• The paper establishes an equivalence between the higher Zariski spectrum of 2-rings and their associated Balmer spectra.
• It utilizes tensor-triangular techniques and categorical localizations to model stable homotopy categories and classify thick tensor ideals.
• The paper demonstrates a Mayer–Vietoris descent formalism that reconstructs global sections from local data within higher-categorical frameworks.

Higher Zariski Geometry

Introduction

The paper "Higher Zariski Geometry" explores the interplay between the traditional Zariski geometry of commutative ring spectra and its categorification via stable $\infty$-categories, referred to as 2-rings. This new framework, termed "Higher Zariski Geometry," is rooted in the development of a Zariski topology for 2-rings, encapsulating classical notions within a higher-categorical context.

Tensor-Triangular Geometry and Zariski Spectra

The classical Zariski geometry studies a commutative ring through its spectrum, a locally ringed space corresponding to prime ideals. Similarly, the spectrum of a 2-ring, built using the Balmer spectrum of tensor-triangulated categories, offers insight into the structure of stable homotopy theories. Balmer's approach utilizes categorical localizations to model these spectra, facilitating the classification of thick tensor ideals through a lattice-theoretic method, yielding insights into the algebraic and geometric properties of these spaces.

2-Rings and Higher Zariski Foundations

A commutative 2-ring or 2CAlg is a small, stable, symmetric monoidal idempotent-complete $\infty$-category. The paper investigates the geometry associated with 2-rings by defining a set of admissible morphisms, analogous to localizations in classical settings. This extends the geometric framework to encompass a broader spectrum of algebraic structures, connecting to the homotopy categories associated with tensor-triangular geometry.

Key Results and Affine Structures

The foundational result establishes an equivalence between the absolute higher Zariski spectrum of a 2-ring and its sheaf of 2-rings on the Balmer spectrum. This intricate correspondence between the spectral framework and tt-geometry (tensor-triangular geometry) reveals a categorical affinity that enhances the understanding of local-to-global principles in stable homotopy theory. Additionally, the paper demonstrates that rigid 2-rings embeddable in locally ringed topoi result in a robust Zariski descent formalism analogous to classical algebraic settings.

Descent and Gluing Phenomena

Key descent results demonstrated include the Mayer–Vietoris sequence for mapping objects, enabling the reconstruction of global sections from local data within this geometric framework. By extending these classical descent methodologies to the 2-rings, the paper lays down a foundational theory for coherently patching local data in expanded algebraic contexts.

Comparisons and Implications

The paper also discusses comparisons to classical Zariski and Dirac geometries, establishing a bridge between these frameworks and demonstrating that the absolute spectrum of certain categories can reflect known geometric objects, thereby extending classical theorems like Thomason's reconstruction in new categorical directions.

Conclusion

"Higher Zariski Geometry" introduces a rich interplay between stable higher-categorical algebra and classical geometric frameworks, providing a coherent method for pursuing descent, localizations, and dualities in enriched algebraic environments. This development opens pathways for further exploration into the algebraic geometry of 2-rings, with implications spanning the classification problems in homotopy theory and beyond.