Tensor triangular geometry of filtered objects and sheaves

TL;DR

This paper computes the Balmer spectra of compact objects in tensor triangulated categories involving filtered or graded objects and sheaves, using an $mbda$-categorical and point-free approach.

Contribution

It introduces an $mbda$-categorical framework and a point-free method to compute Balmer spectra for filtered and sheaf-valued tensor triangulated categories.

Findings

Computed Balmer spectra for filtered derived categories of schemes

Analyzed spectra of homotopy categories of filtered spectra

Developed an $mbda$-categorical and point-free computational approach

Abstract

We compute the the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. Notable examples include the filtered derived category of a scheme as well as the homotopy category of filtered spectra. We use an $\infty$-categorical method to properly formulate and deal with the problem. Our computations are based on a point-free approach, so that distributive lattices and semilattices are used as key tools. In the appendix, we prove that the $\infty$-topos of hypercomplete sheaves on an $\infty$-site is recovered from a basis, which may be of independent interest.