Noncommutative geometry through monoidal categories I

TL;DR

This paper develops a noncommutative geometric framework using monoidal categories, extending classical geometric concepts and invariants to noncommutative settings, and revealing deep connections between commutative and noncommutative geometry.

Contribution

It introduces a noncommutative algebraic geometry based on monoidal categories, providing geometric interpretations for noncommutative constructions and extending classical invariants to this new setting.

Findings

Noncommutative geometry can be modeled via monoidal categories of quasi-coherent sheaves.

Classical geometric concepts like Morita equivalences are given geometric meaning in the noncommutative context.

Homological invariants such as Hochschild and cyclic homology are derived from the constructed cyclic objects.

Abstract

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way.