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If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for $\mathcal{F}$, then the cover's Čech cohomology computes $H^p(X,\mathcal{F})$. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

The complement $Y$ is covered by the $m$ principal affine opens $D(f_1),\dots ,D(f_m)$. Every finite intersection is again a principal affine open, so this is an acyclic cover for ${\mathcal{O}}_Y$. Its Čech cochain complex has no degree-$m$ term because there is no intersection of $m+1$ distinct cover members. Therefore

Translate $p$ to the origin. The complement ${\mathbb{A}}_k^3\setminus \{0\}$ has the affine cover $D(x),D(y),D(z)$, and its second Čech cohomology isThe class of $x^{-1}{y}^{-1}{z}^{-1}$ is nonzero, so $H^2({\mathbb{A}}_k^3\setminus \{p\},\mathcal{O})\ne 0$. On the other hand, ${\mathbb{A}}_k^3\setminus \ell =D(x)\cup D(y)$, and the first part with $m=2$ gives $H^2({\mathbb{A}}_k^3\setminus \ell ,\mathcal{O})=0$. Since sheaf cohomology is invariant under scheme isomorphism, the two complements are not isomorphic.

Write $A=k[x,y]$ and $\mathfrak{m}=(x,y)$. Since $f(0,0)\ne 0$, the principal open subscheme $D(f)=\mathrm{Spec}{A}_f$ contains $\mathfrak{m}$, andCover $V$ by the two affine opens $D(xf)$ and $D(yf)$, whose intersection is $D(xyf)$. The degree-zero part of the resulting Čech cohomology complex givesinside the fraction field of $A$. The last equality follows because $A_f$ is a unique factorization domain and a rational function regular after localizing at both $x$ and $y$ has no possible prime factor left in its denominator.

The same affine cover is acyclic, so its degree-one Čech group computes sheaf cohomology and givesBefore localizing at $f$, the quotienthas the $k$-basisWriting $f=c+h$ with $c=f(0,0)\in k^{\times }$ and $h\in (x,y)$, multiplication by $h$ is locally nilpotent on $Q$: for each negative monomial, a sufficiently high power of $(x,y)$ moves every term into $A_x+A_y$. Hence $c+h$ acts invertibly on $Q$ by a finite geometric series on each element. Localizing at $f$ therefore leaves $Q$ unchanged, andThe displayed infinite basis proves that this vector space is infinite-dimensional.