Deformation of diagrams

TL;DR

This paper develops a deformation theory for diagrams of modules in non-commutative schemes, introducing entanglement among points and tangent directions, and establishing a foundation for non-commutative algebraic geometry.

Contribution

It defines deformation theory for diagrams in non-commutative schemes, incorporating entanglement and tangent directions, and demonstrates its application through simple geometric examples.

Findings

Introduces a new deformation framework for non-commutative diagrams.

Shows how entanglement affects the structure of non-commutative spaces.

Provides examples illustrating the theory's implications.

Abstract

In this this paper we introduce entanglement among the points in a non-commutative scheme, in addition to the tangent directions. A diagram of $A$-modules is a pair $\uc=(|\uc|,\Gamma)$ where $|\uc|={V_1,...,V_r}$ is a set of $A$-modules, and $\Gamma=\{\gamma_{ij}(l)\}$ is a set of $A$-module homomorphisms $\gamma_{ij}(l):V_i\rightarrow V_j$, seen as the 0'th order tangent directions. This concludes the discussion on non-commutative schemes by defining the deformation theory for diagrams, making these the fundamental points of the non-commutative algebraic geometry, which means that the construction of non-commutative schemes is a closure operation. Two simple examples of the theory are given: The space of a line and a point, which is a non-commutative but untangled example, and the space of a line and a point on the line, in which the condition of the point on the line gives an entanglement between the point and the line.