<p>Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space by constructing functors assigning to every object in the category a corresponding sheaf of modules over the structure sheaf of the spectrum. Taking the support of these associated sheaves recovers a notion of support based on local categories. 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 the tt-category satisfies a scheme-like property, then the sheaves associated to objects are 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. </p>