On the module of first and second order differentials of R⊗_k S

Abstract

Let R and S be k-algebras with characteristic zero. Let Ω_k^1 (R⊗_k S) and Ω_k^2 (R⊗_k S) are first and second order universal differential modules over R⊗_k S, respectively. The main result of this paper asserts that in which cases Ω_k^1 (R⊗_k S) and Ω_k^2 (R⊗_k S) can be free modules by using symmetric derivation.