Given a commutative ring $R$, what are relations between w.gldim$(R)$ and w.gldim$(R[[x]])$ (gldim$(R)$ and gldim$(R[[x]])$)?
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Let $R$ be commutative.
Now let $R$ be Noetherian. Hence $R[[X]]$ is Noetherian and since global and weakglobal dimension agree for Noetherian rings, we obtain: The first result is Theorem 1.12 of the paper and the second is Lemma 1 of Jondrup, Small: Power Series over coherent rings, Math. Scand. 35(1974), 2124. 
