The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull Dimension

TL;DR

This paper explores the topological properties of the injective spectrum of right noetherian rings, linking it to Krull dimension and providing computations for specific examples.

Contribution

It establishes foundational topological results and functoriality for the injective spectrum of right noetherian rings, connecting it to Krull dimension.

Findings

Topological properties of injective spectra are characterized.

Functoriality of the injective spectrum is demonstrated.

Examples of injective spectra are explicitly computed.

Abstract

The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples.