tt-geometry of filtered modules

TL;DR

This paper computes the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring, leading to a classification of their thick tensor ideals using two different proof methods.

Contribution

It introduces a new analysis of filtered modules' perfect complexes and provides two proofs, including a direct approach with new tools, expanding understanding in tensor triangular geometry.

Findings

Classification of thick tensor ideals for filtered modules

Reduction to graded modules for spectrum computation

Development of new tools for direct proof

Abstract

We compute the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring, and deduce a classification of the thick tensor ideals. We give two proofs: one by reducing to perfect complexes of graded modules which have already been studied in the literature, and one more direct for which we develop some useful tools.