  We define the spectrum of a tensor triangulated category $K$ as the set of
so-called prime ideals, endowed with a suitable topology. In this very
generality, the spectrum is the universal space in which one can define
supports for objects of $K$. This construction is functorial with respect to
all tensor triangulated functors. Several elementary properties of schemes hold
for such spaces, e.g. the existence of generic points and some
quasi-compactness. Locally trivial morphisms are proved to be nilpotent. We
establish in complete generality a classification of thick tensor-ideal
subcategories in terms of arbitrary unions of closed subsets with quasi-compact
complements (Thomason's theorem for schemes, mutatis mutandis). We also equip
this spectrum with a sheaf of rings, turning it into a locally ringed space. We
compute examples and show that our spectrum unifies the schemes of algebraic
geometry and the support varieties of modular representation theory.
