-linear derivations of a (not necessarily commutative) associative unital -algebra with values in -bimodules are often considered: these are -linear maps satisfying the Leibniz identity: . One would like to construct a universal construction of that kind.
The triple is then called a differential calculus over , provided is in the -linear span of all expressions of the form where . Morphisms of differential calculi over are straightforward to define. A universal object in that category is the universal differential calculus and in that case is called the universal differential envelope of . It is constructed as the quotient of the tensor algebra modulo the ideal generated by the (“augmentation”) kernel of the multiplication map ; then .
This is a bit confusing. The subbimodule should rather be the degree-1 component of (see Bourbaki, Algèbre III, paragraph Problème universel pour les dérivations: cas non commutatif). —Benoit Jubin
Eric: I’m probably confused, but is degree-0 and is degree-1.
Benoit Jubin: I agree. To me, we should have , whereas the article currently proposes .
The bimodule is in fact a -algebra, generated by degree- elements (in ); differential can be naturally extended along the inclusion to a differential satisfying the Leibniz rule, by setting , , where are forms and is homogeneous.