Module des diff{\'e}rentielles

Abstract

For any object x in a category C it is possible to define the category of Beck modules over x as the category Ab(C/x) of abelian group objects in the category C/x. We can deduce from this construction, at least for any locally presentable category, the notion of cotangent module or module of differentials $\Omega$x of x in Ab(C/x).In the case of the category Algk of commutative k-algebras over a ring k, the category of Beck modules Ab(Algk/A) over a k-algebra A is equivalent to the category ModA of A-modules and the cotangent module is equal to the module of K{\"a}hler differentials of A.The aim of this article is to prove for any locally presentable category some results which generalize the classical properties of modules of K{\"a}hler differentials.