Valuative dimension can be computed on an affine cover

The supremum over the cover is at most ${\dim }_{v}X$ by [F1]. Conversely, take an arbitrary affine open $U=\mathrm{Spec}A$, a prime $\mathfrak{p}\subset A$ and a valuation overring $V$ of $A\mathrm{/}\mathfrak{p}$. The induced map $\mathrm{Spec}V\to U$ has closed point $x$ equal to the contraction of the maximal ideal of $V$. Choose a cover member $U_i=\mathrm{Spec}B$ containing $x$. Every point of $\mathrm{Spec}V$ specializes to its closed point, and an open containing a specialization contains its generization. Hence the map has image in $U_i$ and factors through it as a scheme morphism.

The corresponding homomorphism $B\to V$ has prime kernel $\mathfrak{r}$, since $V$ is a domain. The generic point of $\mathrm{Spec}V$ maps to the point represented by $\mathfrak{p}$ in $U$ and by $\mathfrak{r}$ in $U_i$; therefore the fraction fields of $A\mathrm{/}\mathfrak{p}$ and $B\mathrm{/}\mathfrak{r}$ identify with the residue field of that common point. Under this identification $V$ contains $B\mathrm{/}\mathfrak{r}$, so it is a valuation overring of that quotient. Thus $\dim V\le {\dim }_{v}B\le {\sup }_i{\dim }_v{A}_i$. Taking both suprema in [F1] proves the reverse inequality. The empty-cover case is immediate.