Noncommutative Gelfand Duality: the algebraic case

TL;DR

This paper introduces a new framework for non-commutative Gelfand duality using derived algebraic geometry, establishing an anti-equivalence between rings and a subcategory of pre-ringed sites, with implications for quantum spacetime geometry.

Contribution

It develops a novel notion of spectrum for non-commutative rings that generalizes the Grothendieck spectrum and aligns with classical cases for finitely generated algebras over complex numbers.

Findings

Category of rings is anti-equivalent to a subcategory of pre-ringed sites

New notion of spectrum compatible with classical Grothendieck spectrum

Framework aims to study geometric properties of quantum spacetimes

Abstract

The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by Grothendieck's work on commutative rings. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. This work aims to represent the starting point for a rigorous study of geometric properties of quantum spacetimes.