Derivation modules of free joins and $m$-adic completions of algebras

Abstract:

A free commutative join of subalgebras corresponds to a direct sum of submodules in a universal derivation module. In particular, indeterminates of a polynomial ring correspond to elements of a linearly independent set in a universal derivation module. As an application, a simple proof of the uniqueness of cardinalities of indeterminates of a polynomial ring can be obtained by using that of linear bases of a free module over a commutative ring. Similar observations are made for $\mathfrak {m}$-adic completions of algebras and their derivation modules. Also, the module of linear differential forms of an $\mathfrak {m}$-adic completion of an algebra is studied.