Higher Zariski Geometry

TL;DR

This paper develops a higher-dimensional Zariski geometry framework for 2-rings within $mbda$-topoi, connecting classical algebraic geometry with modern homotopical and categorical structures, and extends key conjectures to this setting.

Contribution

It introduces a higher Zariski topology and spectrum for 2-rings, linking them to classical spectra and embedding rigid 2-rings into $mbda$-topoi.

Findings

The spectrum of a 2-ring recovers the Balmer spectrum of its homotopy category.

Constructs a Zariski topology and spectrum for 2-rings within $mbda$-topoi.

Proves a stalk-locality principle for the telescope conjecture in the rigid setting.

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 a ``stalk-locality principle'' for the telescope conjecture in the rigid setting, extending earlier work of Hrbek.