Given a commutative ring $R$, what are relations between w.gldim$(R)$ and w.gldim$(R[[x]])$ (gldim$(R)$ and gldim$(R[[x]])$)?



The global (weak) dimension of formal power series rings
Let $R$ be commutative. 


1. If $R$ is Noetherian, then $\text{gl.dim}(R[[X]]) = 1 + \text{gl.dim}(R)$.
2. If $R[[X]]$ is coherent, then $\text{w-gl.dim}(R[[X]]) = 1 + \text{w-gl.dim}(R)$.


Now let $R$ be Noetherian. Hence $R[[X]]$ is Noetherian and since global and weak-global dimension agree for Noetherian rings, we obtain:  

$$\text{gl.dim}(R[[X]]) = 1 + \text{gl.dim}(R) = \text{w-gl.dim}(R[[X]]).$$


The first result is Theorem 1.12 of the paper 


[Auslander, Buchsbaum: Homological dimension in Noetherian rings, Trans. Amer. Math. Soc. 88(1958),194-206](http://www.ams.org/journals/tran/1958-088-01/S0002-9947-1958-0096720-1/S0002-9947-1958-0096720-1.pdf)  


and the second is Lemma 1 of 


[Jondrup, Small: Power Series over coherent rings, Math. Scand. 35(1974), 21-24](http://www.digizeitschriften.de/en/dms/toc/?PPN=PPN35397434X_0035).


