The Spectrum of Stable Infinity Categories with Actions

Abstract: In the 2000s, Balmer constructed a ringed space from a triangulated category with a tensor structure. It provides a beautiful classification of tensor ideals. However, some categories, such as the bounded derived category of coherent sheaves or the triangulated category of singularities of a scheme, do not have canonical tensor structures. We constructed a ringed space from a stable infinity category with action as an analog. The aforementioned examples have canonical action of the category of perfect complexes, and their associated spaces contain information about the singularities of the scheme. Recently, Matsui introduced a ringed space from a triangulated category. However, it can be huge and hard to calculate. Our approach describes subspaces of the Matsui spectrum, which is more manageable due to their fiber decomposition.