3 yet another typo
2 typo

1

# FInitely-affine morphisms; cohomological dimension of schemes

Let $f\colon X\to U$ be a morphism of Noetherian schemes such that the scheme $U$ is affine and the scheme $X$ is separated and, e.g., quasi-projective over affine. Let $U=\bigcup_\alpha U_\alpha$ be an affine open covering of $U$. Suppose that for each $\alpha$ the full preimage $f^{-1}(U_\alpha)$ can be covered by at most $n$ affine open subschemes. Is there any bound on the number of affine open subschemes covering $X$?

Of course, the bound is supposed to depend only on the number $n$ and not on the number of open subschemes in the covering $U_\alpha$ (or otherwise it is not interesting). It is well-known that if $n=1$ then the morphism $f$ is affine and the scheme $X$ is affine. What happens for $n\ge2$?

One approach to this question would be to notice that one has $R^if_*(\mathcal F)=0$ for any quasi-coherent sheaf $\mathcal F$ on $X$ and all $i\ge n$, hence also $H^i(X,\mathcal F)=0$ for all $i\ge n$. Is there any way to obtain a bound on the number of open affines covering $X$ from this finiteness of cohomological dimension?

The only reference I was able to find is Hartshorne "Cohomological dimension of algebraic varieties", Annals of Math. 88, 1968 (referred to from Exercise III.4.8 in Hartshorne's "Algebraic geometry"), but this doesn't seem to answer my question.