2024/11/18 by Federico Bambozzi, Bambozzi, Federico, Matteo Capoferri +3
Mathematics · #16E35 #18F20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 14A22 #Quantum Algebra (math.QA) #Secondary 14A30
paper · pdf · doi:10.48550/arxiv.2411.11816
openalex publication_date 2024/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.