Finite valuative dimension for noetherian and finite type schemes

Let $A$ be a noetherian ring. For each prime $\mathfrak{p}$, [F2] makes $A\mathrm{/}\mathfrak{p}$ a noetherian domain of dimension at most $\dim A$. By [F1], its valuative dimension equals its Krull dimension. Taking the supremum over prime quotients gives ${\dim }_{v}A\le \dim A$, including the zero-ring convention.

For an affine open $U=\mathrm{Spec}A$ of a scheme $X$ of dimension $d$, a strict chain of primes in $A$ gives a strict chain of irreducible closed subsets of $U$ in reverse order. Taking their closures in $X$ gives a strict chain of irreducible closed subsets: intersecting each closure with the open set $U$ recovers the original closed subset. Therefore $\dim A\le d$. If $X$ is noetherian, choose its finite noetherian affine cover. Step 1.1 bounds each cover member's valuative dimension by $d$, and [F4] gives ${\dim }_{v}X\le d$. The empty scheme satisfies the same assertion by convention.

If $X$ is finite type over $k$, choose a finite affine cover by spectra of finitely generated $k$-algebras. By [F3] each coordinate ring is noetherian of finite dimension, so step 1.1 gives a finite bound for its valuative dimension. The maximum of these finitely many bounds is finite, and [F4] gives the result. The empty cover gives dimension $-1$.