The spectrum of stable infinity categories with actions

Abstract We introduce the relative Matsui spectrum, a new invariant associated with a stable infinity category equipped with an action. This construction generalizes both Balmer’s tensor triangular spectrum and Matsui’s triangular spectrum and provides a unified framework for classifying thick submodules. We establish its fundamental properties, including universality, comparisons with existing spectra and Stevenson’s support, fiberwise decomposition, and descent. Applications show that the relative Matsui spectrum recovers the underlying classical geometric spaces from categorical data in various settings: categories of perfect complexes of schemes, twisted derived categories, categories of singularities, and derived matrix factorization categories. Thus, the relative Matsui spectrum extends tensor triangular geometry beyond settings with a global tensor structure, while preserving geometric intuition.