Given a commutative ring R, what are relations between w.gldm(R) and w.gldm(R[[x]]) ( gldm (R) and gldm(R[[x]]) )?
Let $R$ be commutative.
Now let $R$ be Noetherian. Hence $R[[X]]$ is Noetherian and since global and weak-global dimension agree for Noetherian rings, we obtain:
The first result is Theorem 1.12 of the paper
and the second is Lemma 1 of