The formal spectrum of a tensor-triangulated category

TL;DR

This paper introduces the formal spectrum of a tensor-triangulated category associated with a Thomason subset, exploring its properties and computing examples across algebraic geometry and homotopy theory.

Contribution

It defines the formal spectrum for tensor-triangulated categories and analyzes its properties with computations in various mathematical contexts.

Findings

Established basic properties of the formal spectrum.

Computed examples in algebraic geometry and homotopy theory.

Connected formal spectrum to existing mathematical frameworks.

Abstract

To any essentially small tensor-triangulated category $\mathcal{K}$ and Thomason subset $Y \subseteq \mathrm{Spc}(\mathcal{K})$ we associate a ringed space $(\mathrm{Spf}(\mathcal{K},Y), \mathcal{O}_{\mathrm{Spf}(\mathcal{K},Y)}),$ called the formal spectrum of $(\mathcal{K},Y)$. We establish basic properties of this construction and compute it in several examples from algebraic geometry, chromatic homotopy theory, equivariant homotopy theory, and modular representation theory.