The action of the Cremona group on the non-commutative ring

TL;DR

This paper constructs a non-commutative ring analog of the field of two variables and demonstrates that the Cremona group embeds into its outer automorphism group, linking algebraic geometry and non-commutative algebra.

Contribution

It introduces a novel non-commutative ring model and proves the embedding of the Cremona group into its outer automorphisms, providing both a technical and a conceptual proof.

Findings

Cremona group embeds into outer automorphisms of the non-commutative ring

Provides a new non-commutative algebraic framework related to algebraic geometry

Offers two proofs, one technical and one conceptual, connecting derived categories and non-commutative rings

Abstract

The Cremona group acts on the field of two independent commutative variables over complex numbers. We provide a non-commutative ring that is an analog of non-commutative field of two independent variables and prove that the Cremona group embeds in the group of outer automorphisms of this ring. First proof of this result is technical, the second one is conceptual and gives a way to obtain non-commutative rings from the bounded derived categories of coherent sheaves.