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If a separated scheme has an acyclic affine cover by $r$ open sets, then every quasi-coherent sheaf on it has vanishing sheaf cohomology in degrees at least $r$. The corresponding Čech cochain complex has no terms in those degrees.

Take the projective line with a doubled point over $\mathbb{C}$, obtained by gluing two copies of ${\mathbb{P}}_{\mathbb{C}}^1$ along the complement of one point. After every base change of a morphism of schemes, a closed subset has closed image from each of the two projective-line charts, so its total image, the union of those two images, is closed. The structure morphism is therefore universally closed. The two doubled points have no disjoint neighborhoods, so the scheme is not separated and hence is not proper.

The coincidence locus of two scheme morphisms is the fibre productThe diagonal morphism ${\Delta }_Y$ is a locally closed immersion, and this property is stable under base change of a morphism of schemes, so $E\hookrightarrow X$ is locally closed. The universal property of a fibre product says that a morphism $T\to X$ factors through $E$ exactly when $f{|}_T=g{|}_T$. Thus $E$ is the largest locally closed subscheme on which they coincide. If $Y$ is separated, ${\Delta }_Y$ is a closed immersion, and its base change $E\hookrightarrow X$ is closed.

Choose a finite affine cover $\mathcal{U}=(U_j)$ of the Noetherian scheme $X$. Because $X$ is separated, every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the closed immersion is also affine. By the definition of the direct image sheaf,The Čech complexes for $i_{\ast }\mathcal{F}$ on $\mathcal{U}$ and for $\mathcal{F}$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant quasi-coherent sheaves, so the acyclic cover theorem givesfor every $q$. This is cohomology under a closed immersion.

For $X={\mathbb{P}}_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The cohomological dimension bound from an affine cover therefore yields

A group scheme over $k$ is a $k$-scheme $G$ equipped with multiplication $m:G{\times }_{k}G\to G$, identity $e:\mathrm{Spec}k\to G$, and inversion $j:G\to G$ satisfying the associativity, identity, and inverse diagrams.

ConsiderThe diagonal morphism is ${\Delta }_{G/k}=q^{-1}(e)$. For a finite-type $k$-scheme the rational identity point is closed, so its inverse image is closed. Thus the diagonal is a closed immersion and every such group scheme over a field is a separated scheme.

In characteristic $p>0$, the infinitesimal additive groupis a nonreduced group scheme. Its comultiplication is $t\mapsto t\otimes 1+1\otimes t$, which is well defined because $(t\otimes 1+1\otimes t{)}^p=t^p\otimes 1+1\otimes t^p$.