A brief note on the existence of connections and covariant derivatives on modules

Abstract: In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate o...

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Autores principales:	Rojas,Jacqueline, Mendoza,Ramón
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2017
Materias:	Connection covariant derivative projective module
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200225
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spelling	oai:scielo:S0716-091720170002002252017-07-06A brief note on the existence of connections and covariant derivatives on modulesRojas,JacquelineMendoza,Ramón Connection covariant derivative projective module Abstract: In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate our attention on a K-algebra A which is commutative, and use the Kähler differentials module, , to define connection (see Subsection 2.1). In this context, it is verified that the existence of connections implies the existence of covariant derivatives (cf. Prop. 2.3), and that every projective module admits a connection (cf. Prop. 2.5). Next (in Section 3), we focus our attention in the discussion of some counterexamples comparing these two notions. In fact, it is known that these two notions are equivalent when we consider regular K -algebras of finite type (see 18, Prop. 4.2). As well as, that there exists a connection on M if, and only if, the Atiyah-Kodaira-Spencer class of M, c(M), is zero (see 17 , Prop. 4.3). Finally, we take into account the case where A is (not necessarily commutative) and it is used the bimodule, , of noncommutative differentials introduces by Connes 9) ,10 in place of Kähler differentials to define a connection. In this case, it is proven that a module admits such connection if, and only if, it is a projective module (see 25, Theorem 5.2).info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.36 n.2 20172017-06-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200225en10.4067/S0716-09172017000200225
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Connection covariant derivative projective module
spellingShingle	Connection covariant derivative projective module Rojas,Jacqueline Mendoza,Ramón A brief note on the existence of connections and covariant derivatives on modules
description	Abstract: In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate our attention on a K-algebra A which is commutative, and use the Kähler differentials module, , to define connection (see Subsection 2.1). In this context, it is verified that the existence of connections implies the existence of covariant derivatives (cf. Prop. 2.3), and that every projective module admits a connection (cf. Prop. 2.5). Next (in Section 3), we focus our attention in the discussion of some counterexamples comparing these two notions. In fact, it is known that these two notions are equivalent when we consider regular K -algebras of finite type (see 18, Prop. 4.2). As well as, that there exists a connection on M if, and only if, the Atiyah-Kodaira-Spencer class of M, c(M), is zero (see 17 , Prop. 4.3). Finally, we take into account the case where A is (not necessarily commutative) and it is used the bimodule, , of noncommutative differentials introduces by Connes 9) ,10 in place of Kähler differentials to define a connection. In this case, it is proven that a module admits such connection if, and only if, it is a projective module (see 25, Theorem 5.2).
author	Rojas,Jacqueline Mendoza,Ramón
author_facet	Rojas,Jacqueline Mendoza,Ramón
author_sort	Rojas,Jacqueline
title	A brief note on the existence of connections and covariant derivatives on modules
title_short	A brief note on the existence of connections and covariant derivatives on modules
title_full	A brief note on the existence of connections and covariant derivatives on modules
title_fullStr	A brief note on the existence of connections and covariant derivatives on modules
title_full_unstemmed	A brief note on the existence of connections and covariant derivatives on modules
title_sort	brief note on the existence of connections and covariant derivatives on modules
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2017
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200225
work_keys_str_mv	AT rojasjacqueline abriefnoteontheexistenceofconnectionsandcovariantderivativesonmodules AT mendozaramon abriefnoteontheexistenceofconnectionsandcovariantderivativesonmodules AT rojasjacqueline briefnoteontheexistenceofconnectionsandcovariantderivativesonmodules AT mendozaramon briefnoteontheexistenceofconnectionsandcovariantderivativesonmodules
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A brief note on the existence of connections and covariant derivatives on modules

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