On the opposite of the category of rings

TL;DR

This paper introduces a contravariant functor from rings to ringed spaces, extending the classical Spec construction to noncommutative rings, and explores their geometric and sheaf-theoretic properties.

Contribution

It defines NCSpec as a noncommutative analogue of Spec, providing a new framework for associating geometric spaces to rings beyond the commutative case.

Findings

NCSpec(R) generalizes Spec(R) for noncommutative rings

Spaces NCSpec(R) can be glued from local pieces

Quasicoherent sheaves on NCSpec(R) are studied and computed

Abstract

We define a faithful contravariant functor NCSpec from the category of rings to the category of ringed spaces, and show that if R is a commutative ring then NCSpec(R) may be viewed as a completion of Spec(R) in an appropriate sense. We then explain how the spaces NCSpec(R) may be glued, and study quasicoherent sheaves on them. As an example, we compute the category of quasicoherent sheaves on a space constructed from a skew-polynomial ring R by an analogue of the Proj construction.