= Solution
Suppose $X=U_1\cup\cdots\cup U_n$ with each $U_i$ an <affine open subscheme>. Every <affine scheme> is a <quasi-compact topological space>: a cover by <principal open subsets> $D(f_\lambda)$ means that the $f_\lambda$ generate the <unit ideal>; an expression $1=\sum_{j=1}^r a_jf_{\lambda_j}$ gives a finite subcover. Every <open cover> has a refinement by <principal open subsets>, so this proves the general covering property.
Restrict an arbitrary <open cover> of $X$ to each $U_i$ and choose a finite subcover there. The union of these finitely many finite families covers $X$. Thus $X$ is a <quasi-compact scheme>.
Now assume $X$ is a <separated scheme>. By (c), every finite intersection of the $U_i$ is an <affine scheme>, and hence has vanishing positive <sheaf cohomology> for the <structure sheaf>. This <affine open cover> therefore computes <sheaf cohomology> using its <Čech cochain complex>. Its degree $p$ term is
$$
\check C^p=\prod_{1\leq i_0<\cdots<i_p\leq n}\Gamma(U_{i_0}\cap\cdots\cap U_{i_p},\mathcal O_X).
$$
There are no strictly increasing $(p+1)$-tuples when $p\geq n$, so those <Čech cochain groups> vanish. Consequently
$$
\boxed{H^m(X,\mathcal O_X)=0\qquad(m\geq n).}
$$
The same argument works for every <quasi-coherent sheaf>. The relevant comparison requires the acyclicity of the finite intersections; the bare assertion that arbitrary <Čech cohomology> equals <sheaf cohomology> would be too broad.
Back to article page
