Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Abstract

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of $\mathbb{Z}$. We also give a first discussion of the spectrum in two new examples, namely equivariant $\mathit{KK}$–theory and stable ${\mathbb{A}}^1$–homotopy theory.