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 ε 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 D K (A, ε) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module D K (A, ε) with the A-module Hom A (Ω K (A), ε). where A, in addition of being associative and unital, is assumed to be commutative, and Ω K (A) denotes the A-module of germs of Kähler differentials. Finally, we expound on functoriality of Kähler differentials.