Even more spectra: tensor triangular comparison maps via graded commutative 2-rings

TL;DR

This paper introduces graded commutative 2-rings as a categorification of graded rings, extending Balmer's comparison maps to tensor-triangulated categories, and computes spectra for derived categories and schemes.

Contribution

It develops the theory of graded commutative 2-rings and generalizes spectrum comparison maps in tensor-triangulated categories.

Findings

Computed the spectrum of the derived category of perfect complexes over graded rings.

Embedded schemes with ample line bundles into spectra of graded commutative 2-rings.

Established a systematic framework for categorified spectra in tensor-triangulated geometry.

Abstract

We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer's comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.