  As an extension of an author's previous paper, we prove the Gersten-type
conjecture for the mod $p$ \'{e}tale motivic cohomology over a local ring of
mixed characteristic $(0, p)$. We also prove the $\mathbb{P}^{1}$-homotopy
invariance for the generalized Brauer group which is a genelization of a result
due to Saltman in mixed characteristic cases.
