Associated sheaf functors in tt-geometry

TL;DR

This paper explores the geometric structure of tensor triangulated categories by constructing sheaves and support theories, establishing conditions under which these notions align with classical schemes and cohomology.

Contribution

It introduces a sheaf-theoretic framework for tt-geometry, linking support, sheaves, and cohomology, and compares it with existing support notions.

Findings

Support and sheaf constructions coincide on compact objects.

Sheaves are quasi-coherent under scheme-like conditions.

Sheaves correspond to zeroth cohomology with suitable t-structures.

Abstract

Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space. Specifically we construct functors assigning to every object in the category a corresponding sheaf and a notion of support based upon these sheaves. We compare this support to the usual support in tt-geometry and show that under reasonable conditions they agree on compact objects. We show that when tt-categories satisfy a scheme-like property then the sheaf associated to an object is quasi-coherent, and that in the presence of an appropriate t-structure and affine assumption, this sheaf is in fact the sheaf associated to the object's zeroth cohomology.