# The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories

## TL;DR

This paper extends the study of the injective spectrum of a right noetherian ring by defining a sheaf of rings, exploring sheaves of modules, and linking these to prime torsion theories, enriching the topological and algebraic understanding.

## Contribution

It introduces a sheaf of rings on the injective spectrum and connects sheaves of modules to the original ring, expanding the spectral theory framework.

## Key findings

- Defined a sheaf of rings on the injective spectrum
- Linked sheaves of modules to modules over the original ring
- Proved new results relating topology and torsion theories

## Abstract

This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure sheaf and their relation to modules over the original ring. We then explore links with the spectrum of prime torsion theories developed by Golan and use this torsion-theoretic viewpoint to prove further results about the topology.