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The affine plane with doubled origin is formed by gluing two copies of ${\mathbb{A}}_k^2$ by the identity away from the origin. The overlap is the punctured affine plane, so this scheme is nonseparated and the two-open affine cover is not acyclic for the structure sheaf.

The gluing resembles a non-Hausdorff manifold example, but this node concerns an algebraic scheme and its nonseparatedness, not a real manifold.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

The two affine opens $D(x),D(y)$ cover the punctured affine plane. Their intersection has ring $k[x^{\pm 1},y^{\pm 1}]$. The Čech cohomology in degree one is the quotient by $k[x^{\pm 1},y]+k[x,y^{\pm 1}]$. Its basis consists precisely of the Laurent monomials whose two exponents are negative, so this vector space is infinite-dimensional. Global functions, by contrast, form $k[x,y]$.

For $X=\mathrm{Spec}k[x,y]$ and $Z=V(x,y)$, the only nonzero local cohomology group of the structure sheaf isIt has basis $x^{-a}{y}^{-b}$ for $a,b\ge 1$. The two-standard-open Čech cochain complex of the punctured affine plane proves the formula.

The punctured affine plane $U={\mathbb{A}}_k^2\setminus \{(0,0)\}$ is not an affine scheme. On its cover $D(x)\cup D(y)$, global sections are $k[x,y{]}_x\cap k[x,y{]}_y=k[x,y]$ inside the field of fractions. Coprimality of $x$ and $y$ proves this equality. If $U$ were affine, its inclusion into ${\mathbb{A}}_k^2$ would be an isomorphism by affine scheme reconstruction from global sections, contradicting the omitted point.

The point to retain in the direct-image theorem is quasi-compactness of inverse images of affine opens and of overlaps; no separation assumption has been supplied. The underlying space of a Noetherian scheme is Noetherian, so every open subset of $X$ is quasi-compact. Fix an affine open subscheme $V=\mathrm{Spec}A$ of $Y$, put $W=f^{-1}V$, and choose a finite open cover by affine open subschemes $W={\bigcup }_i{U}_i$. For each pair $i,j$, choose a finite affine cover $(W_{ij\ell }{)}_{\ell }$ of $U_i\cap U_j$. Empty overlaps contribute no terms.

Write $M=\Gamma (W,\mathcal{F})$. The sheaf gluing axiom gives the exact sequencewhere $\delta$ is the difference of the two restrictions to each overlap chart. All terms are $A$-modules through $f$. For $a\in A$, the inverse image of $D(a)$ cuts each $U_i$ and $W_{ij\ell }$ by a principal open subscheme. Because $\mathcal{F}$ is a quasi-coherent sheaf, sections on these smaller affine charts are the corresponding module localizations at $a$. Exactness of localization and its commutation with finite products now identify the localized equalizer with the equalizer for the restricted cover. ThusThe isomorphisms respect restrictions, proving $(f_{\ast }\mathcal{F}){|}_V\cong \tilde{M}$ on the basis of principal open subschemes. Since $V$ was arbitrary, $f_{\ast }\mathcal{F}$ is a quasi-coherent sheaf. The finite covers of overlaps are what allow the proof to work for nonseparated Noetherian schemes; this is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism.

For a quasi-coherent sheaf which is not coherent but has coherent direct image, use $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ andThis infinite negative-twist sum with zero global sections is quasi-coherent: on each standard affine chart it is the sheaf associated with a direct sum of free rank-one modules. Its stalk at every point, modulo the maximal ideal, is an infinite-dimensional vector space. A finitely generated module would have a finite-dimensional quotient, so $\mathcal{F}$ is not a coherent sheaf.

Nevertheless, $\Gamma ({\mathbb{P}}^1,\mathcal{O}(-1))=0$. This follows from cohomology of twisting sheaves on projective space, or directly by gluing on the two standard affine charts: if $z=t_1/t_0$, a section is a polynomial $p(z)$ on the first chart and $q(z^{-1})$ on the second with $p(z)=z^{-1}q(z^{-1})$, which forces both to vanish. Global sections commute with this direct sum: they are the kernel of the difference map for the two-chart cover, and direct sums commute with that finite equalizer of modules. Hencewhich is coherent on $\mathrm{Spec}k$. Both schemes in this example are Noetherian.

For a finite morphism, such an example is impossible. On an affine open subscheme $V=\mathrm{Spec}A$ of the target, its inverse image is $\mathrm{Spec}B$, with $B$ a finite $A$-module. Write $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$. Its direct image corresponds to $M$ considered as an $A$-module. If the direct image is coherent, $M$ is finitely generated over $A$. The same generators also generate it over $B$, because $A$ acts through $B$. Since $B$ is Noetherian, $\tilde{M}$ is coherent. Conversely, a finite set of $B$-generators combined with a finite set of $A$-generators of $B$ gives finitely many $A$-generators of $M$. Thus coherence reflected by finite direct image gives the stronger equivalence

Finally, let $Y=\mathrm{Spec}k[x,y]$, let $X=Y\setminus \{(x,y)\}$ be the punctured affine plane, and let $j:X\hookrightarrow Y$ be the open immersion. Both are integral schemes, but $j$ is not an isomorphism of schemes because it omits a point. The cover $X=D(x)\cup D(y)$ givesinside the field of fractions $k(x,y)$. Indeed, in a reduced fraction, membership in the first localization forces every denominator factor to be associated to $x$, while membership in the second forces it to be associated to $y$. Unique factorization and coprimality force the denominator to be a unit. By the theorem just proved, $j_{\ast }{\mathcal{O}}_X$ is quasi-coherent on the affine $Y$, hence determined by this module of global sections. The natural map ${\mathcal{O}}_Y\to j_{\ast }{\mathcal{O}}_X$ corresponds to the identity of $k[x,y]$, soa coherent sheaf. This is a concrete case of codimension-two extension of regular functions on a normal variety.

Put $A=k[t_1,t_2]$. The punctured affine plane is covered by the principal open subschemes $D(t_1)$ and $D(t_2)$. Since $A$ is an integral domain, restriction identifies its global sections withThis intersection is $A$: if $a/t_1^r=b/t_2^s$, then $t_2^{s}a=t_1^{r}b$, and unique factorization forces $t_1^r$ to divide $a$. Therefore the inclusion $j:X\hookrightarrow \mathrm{Spec}A$ induces the identity on global regular functions.

If $X$ were an affine scheme, affine scheme reconstruction from global sections would make $j$ an isomorphism of schemes, because its global-section map is an isomorphism. But its image omits the closed point $(t_1,t_2)$. The punctured affine plane is therefore not affine. This is a useful example of global regular functions failing to distinguish a nonaffine open subscheme from its affine ambient space.

Write $A=k[x,y]$, $X=\mathrm{Spec}A$, and $U=X\setminus Z$. Since $A$ is an integral domain, no nonzero global section is supported only at the origin, soThe punctured affine plane has the affine cover $U=D(x)\cup D(y)$. Its Čech cochain complex for the structure sheaf isConsequently

Because $X$ is affine, the higher sheaf cohomology of ${\mathcal{O}}_X$ vanishes. The long exact sequence for local cohomology therefore givesand $H_Z^i(X,{\mathcal{O}}_X)=0$ for $i\ge 3$. The nonzero group has the $k$-basisThis computes the local cohomology of the affine plane supported at the origin in every degree.

Write $A=k[x,y]$. If $\dim Z=1$, its prime ideal has height one and is generated by an irreducible polynomial $f$, because $A$ is a unique factorization domain. Then $U=D(f)$ is a principal open subscheme andIf $\dim Z=0$, its complement has codimension two. Since ${\mathbb{A}}_k^2$ is normal, regular functions extend across that codimension-two subset, givingThis includes the calculation for the punctured affine plane. If $Z={\mathbb{A}}_k^2$ has dimension two, then $U$ is empty and its ring of sections is the zero ring.

A morphism $f:X\to S$ is a separated morphism when its diagonal morphismis a closed immersion.

For the requested example, take the affine plane with doubled origin: glue two copies $U,V\cong {\mathbb{A}}_k^2$ by the identity onThe opens $U$ and $V$ are affine, while their intersection is the punctured affine plane, which is not affine. Indeed, its regular functions are still $k[x,y]$; if it were affine, the canonical map to $\mathrm{Spec}k[x,y]={\mathbb{A}}_k^2$ would be an isomorphism, contrary to the missing origin. The resulting scheme is not separated: in a separated scheme, the intersection of two affine opens is the inverse image of the closed diagonal inside their affine product and is therefore affine.

For the cover $\mathcal{U}=\{X_1,X_2\}$ of the affine plane with doubled origin, the overlap is the punctured affine plane $U$. Since $H^0(U,{\mathcal{O}}_U)=k[x,y]$, the Čech complex beginsThis map is surjective, and the normalized complex has no terms in degrees at least two. Consequently

The Mayer-Vietoris sequence for sheaf cohomology also gives $H^1(Y,{\mathcal{O}}_Y)=0$, but its next part givesPart b with $f=1$ shows that the group on the right is infinite-dimensional. Thus

This does not contradict the acyclic cover theorem. Although $X_1$ and $X_2$ are affine, their intersection $U$ is not acyclic: it has nonzero first structure-sheaf cohomology. Equivalently, this affine cover does not satisfy the theorem's hypotheses; the doubled-origin plane is not a semi-separated scheme.