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 our attention on a k-algebra A which is commutative, and use the Kahler differentials module, Ω¹A/k, 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, Ω¹, of noncommutative differentials introduces by Connes [9], [10] in place of Kahler 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).