The formal spectrum of a tensor-triangulated category

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.

35 pages. Comments welcome! v2: many small corrections and improvements, as well as a new section constructing the sheaf at the \infty-categorical level