Dérivations Et Calcul Différentiel Non Commutatif Ii Dérivations and Noncommutative Differential Calculus Ii

We characterize the derivation d : A→ ΩDer(A) by a universal property introducing a new class of bimodules. Abridged English Version In this Note, A denotes an associative algebra over K = R or C with a unit 1l. In the Note [5], a graded differential algebra ΩDer(A) with Ω 0 Der(A) = A was introduced with the notation ΩD(A). It is one of the aims of this Note to show that, by introducing a new category of bimodules, the derivation d : A→ ΩDer(A) is characterized by a universal property. Before introducing this category of bimodules, we first consider a slightly bigger category of bimodules : we call central bimodule a bimodule such that the corresponding bimodule structure over the center Z(A) of A is induced by a structure of Z(A)-module. The category of central bimodules is a full subcategory of the category of all bimodules and we construct for the corresponding embedding functor a left adjoint M 7→ MZ and a right adjoint M 7→ M . Consequently, the category of central bimodules is stable by taking subbimodules, quotient modules, arbitrary projective limits and arbitrary inductive limits. This category is obviously stable by taking tensor products over Z(A), and consequently also over A. By applying the functor M 7→ MZ to the universal differential calculus on d : A → Ω(A), we produce a universal differential calculus for central bimodules, i.e. a derivation dZ : A → Ω(A)Z where Ω(A)Z is central such that any derivation of A in a central bimodule factorizes through dZ and a unique homomorphism from Ω(A)Z in the bimodule. In the case where A is commutative, a central bimodule is simply a module and Ω(A)Z reduces to the module of Kähler K -differentials ΩK(A). We now introduce the appropriate category of bimodules to deal with d : A→ ΩDer(A). We call diagonal bimodule