Let $f \colon X \to S$ be a proper morphism form a seperated algebraic space $X$ to an affine noetherian scheme $S$.


Given a coherent sheaf $F$ on $X$, we know from Knutson's book, that the finiteness theorem holds: The $R^qf\_\*F$ are finite $\mathcal O\_S$-modules. For complete $S$, we also have the theorem of formal functions.


Now, let us assume that $F$ is flat over $S$. If $X$ were a scheme, we would know that the function on $S$, given by the dimension of the cohomology of $F$ restricted to fibers, is upper semicontinous. Is the same also true for $X$ an algebraic space?


Moreover, if we knew that for some $i \ge 0$ the function $s \mapsto h^i(X\_s, F\_s)$ is constant, than for $X$ a scheme, it would follow that the base change map $R^i f\_\* (F) \otimes k(s) \to H^i(X\_s, F\_s)$ is an isomorphisms. What about algebraic spaces?


many thanks in advance.



Semicontinuity and cohomological flatness for algebraic spaces
This is definitely true and there ought to (but may not) exist a reference. One
way to prove it is just to check that the usual proof for schemes extends. If we
follow Hartshorne for instance the crucial part is Proposition III:12.2 which
can be proven using an étale affine cover instead of an open affine one.


