Universal problem for Kähler differentials in A-modules: Non-commutative and commutative cases

Abstract

Let A be an associative and unital K-algebra sheaf, where K is a commutative ring sheaf, and e an (A, A)-bimodule, that is, a sheaf of (A, A)-bimodules. We construct an (A, A)-bimodulc which is K-isomorphic with the K-module DK(A, e) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module DK(A, e) with the A-module HomA(ΩK(A), e). where A, in addition of being associative and unital, is assumed to be commutative, and ΩK(A) denotes the A-module of germs of Kahler differentials. Finally, we expound on functoriality of Kahler differentials.