 Solution

Write the affine variety as $X=\mathrm{Spec}A$. Its coordinate ring is Noetherian, and a coherent sheaf is $\mathcal{F}=\tilde{M}$ for a finitely generated $A$-module $M$. We first justify the acyclic resolution used to compute its sheaf cohomology.

If $I$ is an injective module, then $\tilde{I}$ is a flasque sheaf. Indeed every open $U$ in this Noetherian scheme has the form $X\setminus V(\mathfrak{a})$ for a finitely generated ideal. Clearing denominators givesHere a homomorphism $h:{\mathfrak{a}}^n\to I$ gives, on a principal open $D(f)$ with $f\in \mathfrak{a}$, the fraction $h(f^n)/f^n$; the fractions agree on overlaps. Conversely choose fractions for a section on a finite principal-open cover. A common power clears their denominators and the finitely many compatibility relations. Multiplication by a further power of $\mathfrak{a}$ kills any elements supported on $V(\mathfrak{a})$, producing a well-defined map on some ${\mathfrak{a}}^n$. Changing that power accounts for the direct limit. This is the sheafification of an injective module is flasque construction. Since $I$ is injective, each map ${\mathfrak{a}}^n\to I$ extends to $A\to I$. Thus every section on every open extends to $X$. Extending globally and restricting again also proves surjectivity for restrictions between arbitrary opens.

Choose an injective module resolution $0\to M\to I^0\to I^1\to \cdots$. Localization is exact, so sheafifying gives a flasque resolution of $\mathcal{F}$. Its global-section complex is exactly $I^{\bullet }$, since $\Gamma (\mathrm{Spec}A,\tilde{I})=I$. This complex is exact in positive degrees, and a flasque resolution computes sheaf cohomology. ConsequentlyThe argument actually proves vanishing of quasi-coherent cohomology on an affine scheme, without finite generation of $M$.

For the intersection assertion, varieties are taken to be separated, as usual. The diagonal morphism $X\to X{\times }_{k}X$ is a closed immersion. Its inverse image over $U{\times }_{k}V$ identifies $U\cap V$ with a closed subvariety of the affine variety $U{\times }_{k}V$. A closed subvariety of an affine variety is affine: its coordinate ring is the ambient coordinate ring modulo its defining ideal. Thus $U\cap V$ is affine. Repeated application gives affine finite intersections. Without separatedness this statement need not hold.

Choose a finite affine open cover $(U_{\alpha })$ of the projective variety $X$. For ordered indices write $U_{{\alpha }_0\cdots {\alpha }_p}=U_{{\alpha }_0}\cap \cdots \cap U_{{\alpha }_p}$. The answer is the cohomology of the Čech cochain complexIn particular $H^0=\ker (C^0\to C^1)$ and, in positive degree,The chart modules $\Gamma (U_{\alpha },\mathcal{F})$, together with their restriction and gluing maps, supply the modules on overlaps by localization. The gluing maps are essential; abstract chart modules alone do not specify a sheaf.

For completeness, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$ and its double complex of Čech cochain complexes on this cover. In the resolution direction, its cohomology is $H^q(U_{{\alpha }_0\cdots {\alpha }_p},\mathcal{F})$, which vanishes for $q>0$ by the affine result. In the cover direction, the augmented Čech cochain complex of a flasque sheaf is exact: sections can be extended from overlaps, successively adjusting a cocycle to eliminate one cover member at a time. Equivalently the augmented sheaf complex is stalkwise contracted by inserting a cover index containing the point. Thus computing the total cohomology in the two directions identifies Čech cohomology with sheaf cohomology, proving the displayed description.