Deformations of Diagrams

DOI10.1007/978-3-642-55361-5_15zbMATH Open1405.16038arXiv1204.3303OpenAlexW35980774MaRDI QIDQ5258845

Publication date: 24 June 2015

Published in: Springer Proceedings in Mathematics & Statistics (Search for Journal in Brave)

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

u c = (| u c |, G a m m a)

where

| u c | = V_{1}, . . ., V_{r}

is a set of

A

-modules, and

G a m m a = g a m m a_{i j} (l)

is a set of

A

-module homomorphisms

g a m m a_{i j} (l) : V_{i} i g h t a r r o w 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.

Full work available at URL: https://arxiv.org/abs/1204.3303

Mathematics Subject Classification ID

Noncommutative algebraic geometry (14A22) Deformations of associative rings (16S80)