Last update: 11 June 2013
Let $A$ be any commutative ring with identity element, and let $X=\text{Spec}\left(A\right)\text{.}$ We recall that, for any $f\in A,$ the 'basic open set' $D\left(f\right)$ is the set of all $x\in X$ such that $f\left(x\right)\ne 0,$ i.e. such that $f\notin {j}_{x}\text{.}$ Let $M$ be any $A\text{module;}$ then we can form the module of fractions ${M}_{f},$ whose elements are all fractions of the form $m/{f}^{n}$ $\text{(}m\in M$ $n$ an integer $\ge 0\text{).}$ ${M}_{f}$ is a module over the ring ${A}_{f}\text{.}$ We may then consider the presheaf $D\left(f\right)\to {M}_{f},$ defined on the basis $\mathcal{B}={D\left(f\right)}_{f\in A}$ of $X\text{.}$ We denote by $\stackrel{\sim}{M}$ the presheaf on $X$ which it determines. Since each ${M}_{f}$ is an ${A}_{f}\text{module,}$ $\stackrel{\sim}{M}$ is an $\stackrel{\sim}{A}\text{Module.}$
Proposition (9.1). $\stackrel{\sim}{M}$ is a sheaf, and hence $\Gamma (D\left(f\right),\stackrel{\sim}{M})={M}_{f}$ for all $f\in A\text{.}$ In particular $\Gamma (X,\stackrel{\sim}{M})=M\text{.}$
Proof.  
Copy of the proof of (5.1) (ii). $\square $ 
Since formation of modules of fractions preserves exactness, it follows that the functor $M\mapsto \stackrel{\sim}{M}$ (from $A\text{modules}$ to $\stackrel{\sim}{A}\text{Modules)}$ is exact. Moreover,
Corollary (9.2). If $M,N$ are $A\text{modules,}$ then ${\text{Hom}}_{A}(M,N)\cong {\text{Hom}}_{\stackrel{\sim}{A}}(\stackrel{\sim}{M},\stackrel{\sim}{N})\text{.}$
Proof.  
$\phi :M\to N$ gives rise to ${\phi}_{f}:{M}_{f}\to {N}_{f}$ for each $f\in A,$ hence to $\stackrel{\sim}{\phi}:\stackrel{\sim}{M}\to \stackrel{\sim}{N}\text{.}$ Hence we have a homomorphism ${\text{Hom}}_{A}(M,N)\to {\text{Hom}}_{\stackrel{\sim}{A}}(\stackrel{\sim}{M},\stackrel{\sim}{N})\text{.}$ Conversely, given $u:\stackrel{\sim}{M}\to \stackrel{\sim}{N}$ we have $u\left(X\right):\stackrel{\sim}{M}\left(X\right)\to \stackrel{\sim}{N}\left(X\right),$ i.e. by (9.1) a homomorphism $M\to N\text{.}$ Hence a map ${\text{Hom}}_{\stackrel{\sim}{A}}(\stackrel{\sim}{M},\stackrel{\sim}{N})\to {\text{Hom}}_{A}(M,N)\text{.}$ Verify that the two maps so defined are inverses of each other. $\square $ 
Theorem (9.3). Let $\mathcal{F}$ be an $A\text{Module.}$ Then the following are equivalent:
(a)  $\mathcal{F}\cong \stackrel{\sim}{M}$ for some $A\text{module}$ $M\text{;}$  
(b)  there exists a finite open covering of $X$ by basic open sets $D\left({f}_{i}\right)$ such that $\mathcal{F}D\left({f}_{i}\right)\cong {\stackrel{\sim}{M}}_{i}$ for some ${A}_{{f}_{i}}\text{module}$ ${M}_{i}$ and each index $i\text{;}$  
(c)  $\mathcal{F}$ is quasicoherent;  
(d) 
$\mathcal{F}$ satisfies the following two conditions:

Proof according to the scheme  
$$\begin{array}{ccccccc}\text{(a)}& \multicolumn{3}{c}{\u27f9}& \text{(b)}& \u27fa& \text{(c).}\\ & \n\u27fa\n& & \n\u27f8\n\\ & & \text{(d)}\end{array}$$
(a) $\Rightarrow $ (b). Take the covering of $X$ consisting of the single set $D\left(1\right)=X\text{.}$ (b) $\Rightarrow $ (c). Since quasicoherence is a local property, it is enough to prove (a) $\Rightarrow $ (c). We have an exact sequence ${A}^{\left(I\right)}\to {A}^{\left(J\right)}\to M\to 0,$ where ${A}^{\left(I\right)},{A}^{\left(J\right)}$ are direct sums of copies of $A\text{;}$ hence, since $M\mapsto \stackrel{\sim}{M}$ is an exact functor, an exact functor, an exact sequence ${\stackrel{\sim}{A}}^{\left(I\right)}\to {\stackrel{\sim}{A}}^{\left(J\right)}\to \stackrel{\sim}{M}\to 0\text{.}$ Hence $\stackrel{\sim}{M}$ is quasicoherent. (c) $\Rightarrow $ (b). Each $x\in X$ has a neighbourhood $D\left(f\right)$ over which $\mathcal{F}D\left(f\right)$ is the cokernel of a homomorphism ${\stackrel{\sim}{A}}^{\left(I\right)}D\left(f\right)\to {\stackrel{\sim}{A}}^{\left(J\right)}D\left(f\right),$ i.e. a homomorphism ${\stackrel{\sim}{A}}_{f}^{\left(I\right)}\to {\stackrel{\sim}{A}}_{f}^{\left(J\right)}\text{.}$ Hence by (9.2) $\mathcal{F}D\left(f\right)=\stackrel{\sim}{N}$ where $N=\text{coker}({A}_{f}^{\left(I\right)}\to {A}_{f}^{\left(J\right)})\text{.}$ Since $X$ is quasicompact, (b) is proved. (a) $\Rightarrow $ (d). If $g\in A$ and $s\in \mathcal{F}\left(D\left(g\right)\right)={M}_{g},$ then $s=m/{g}^{n}$ for some $m\in M$ and some integer $n\ge 0,$ hence $s{g}^{n}=m/1=$ image of $m$ in ${M}_{g},$ i.e. $s{g}^{n}$ is the image of an element of $M=\mathcal{F}\left(X\right)\text{.}$ If $g\in A,$ $t\in M$ and $t/1=0$ in ${M}_{g},$ then $t{g}^{n}=0$ for some integer $n\ge 0,$ from the basic properties of modules of fractions. (b) $\Rightarrow $ (d). We have to show that if each $\mathcal{F}D\left({f}_{i}\right)$ satisfies (d), then so does $\mathcal{F}\text{.}$ Take $\left({d}_{2}\right)$ first. We have then $g\in A,$ $t\in \mathcal{F}\left(X\right)$ and $tD\left(g\right)=0\text{.}$ Then $tD\left(g{f}_{i}\right)=0$ (since $D\left(g{f}_{i}\right)=D\left(f\right)\cap D\left({f}_{i}\right)\text{);}$ hence by $\left({d}_{2}\right)$ applied to $\mathcal{F}D\left({f}_{i}\right)$ there exists an integer ${n}_{i}\ge 0$ such that ${\left({f}_{i}g\right)}^{{n}_{i}}tD\left({f}_{i}\right)=0,$ i.e. ${\left({f}_{i}g\right)}^{{n}_{i}}t=0$ in ${M}_{i}\text{;}$ now ${f}_{i}$ is a unit in ${A}_{{f}_{i}},$ hence ${g}^{{n}_{i}}t=0$ in ${M}_{i}\text{.}$ Let $n$ be the largest of the ${n}_{i},$ then we have ${g}^{n}t=0$ in each $\mathcal{F}D\left({f}_{i}\right),$ hence ${g}^{n}t$ is the zero section of $\mathcal{F}\text{.}$ To prove $\left({d}_{1}\right)\text{:}$ take $g\in A$ and $s\in \mathcal{F}\left(D\left(g\right)\right)\text{.}$ By applying $\left({d}_{1}\right)$ to $\mathcal{F}D\left({f}_{i}\right),$ there exists an integer ${n}_{i}\ge 0$ and an element ${s}_{i}^{\prime}\in \mathcal{F}\left(D\left({f}_{i}\right)\right)$ which extends ${\left({f}_{i}g\right)}^{{n}_{i}}sD\left({f}_{i}g\right)\text{.}$ Since ${f}_{i}$ is a unit in ${A}_{{f}_{i}},$ there exists ${s}_{i}^{\prime}={f}_{i}^{{n}_{i}}{s}_{i},$ and ${s}_{i}$ extends ${g}^{{n}_{i}}sD\left({f}_{i}g\right)\text{;}$ and we may take all the ${n}_{i}$ to be equal, say ${n}_{i}=n\text{.}$ By construction, ${s}_{i}{s}_{j}$ restricted to $D\left({f}_{i}{f}_{j}g\right)$ is zero; now since $\mathcal{F}D\left({f}_{i}\right)={\stackrel{\sim}{M}}_{i},$ it follows that each $\mathcal{F}D\left({f}_{i}\right)\cap D\left({f}_{j}\right)$ satisfies (a) and therefore (d), hence by $\left({d}_{2}\right)$ applied to $\mathcal{F}D\left({f}_{i}\right)\cap D\left({f}_{j}\right)$ there exist integers ${m}_{ij}\ge 0$ such that ${\left({f}_{i}{f}_{j}g\right)}^{{m}_{ij}}({s}_{i}{s}_{j})$ restricted to $D\left({f}_{i}\right)\cap D\left({f}_{j}\right)=D\left({f}_{i}{f}_{j}\right)$ is zero; but ${f}_{i}{f}_{j}$ is a unit in $\mathcal{F}\left(D\left({f}_{i}{f}_{j}\right)\right),$ hence ${g}^{m}({s}_{i}{s}_{j})$ restricted to $D\left({f}_{i}\right)\cap D\left({f}_{j}\right)$ is zero, where $m=\text{max}\left({m}_{ij}\right)\text{.}$ Hence the ${g}^{m}{s}_{i}\in \Gamma (D\left({f}_{i}\right),\mathcal{F})$ are all of them restrictions of a global section $s\prime $ of $\mathcal{F}\text{.}$ This section $s\prime $ is an extension of ${g}^{m+n}s,$ hence $\left({d}_{1}\right)$ is proved. (d) $\Rightarrow $ (a). Let $M=\mathcal{F}\left(X\right)=\Gamma (X,\mathcal{F})\text{.}$ We shall define a homomorphism $u:\stackrel{\sim}{M}\to \mathcal{F}$ and show that it is an isomorphism. For this we must define ${u}_{f}:{M}_{f}\to \mathcal{F}\left(D\left(f\right)\right)$ for each $f\in A,$ satisfying the usual compatibility conditions. Start with the restriction homomorphism $\mathcal{F}\left(X\right)\to \mathcal{F}\left(D\left(f\right)\right),$ i.e. $M\to \mathcal{F}\left(D\left(f\right)\right)\text{.}$ Since $f$ is a unit in ${A}_{f},$ this homomorphism factorizes through ${M}_{f}\text{:}$ $M\u27f6{M}_{f}\stackrel{{u}_{f}}{\u27f6}\mathcal{F}\left(D\left(f\right)\right)\text{.}$ This defines ${u}_{f}\text{.}$ We shall show that $\left({d}_{1}\right)$ implies ${u}_{f}$ surjective, and $\left({d}_{2}\right)$ implies ${u}_{f}$ injective. Let $s$ be any element of $\mathcal{F}\left(D\left(f\right)\right)\text{.}$ Then by $\left({d}_{1}\right)$ ${f}^{n}s$ lifts to a global section of $\mathcal{F},$ for some integer $n\ge 0,$ i.e. ${f}^{n}s$ is in the image of $M,$ hence is in ${u}_{f}\left({M}_{f}\right)\text{.}$ Hence, as $f$ is a unit in ${A}_{f},$ we have $s\in {u}_{f}\left({M}_{f}\right)$ and thus ${u}_{f}$ is surjective. If $z/{f}^{n}\in {M}_{f}$ is such that ${u}_{f}(z/{f}^{n})=0,$ then ${u}_{f}(z/1)=0$ and therefore the restriction of $z\hspace{0.17em}(\in \mathcal{F}\left(X\right))$ to $D\left(f\right)$ is zero; hence by $\left({d}_{2}\right)$ there exists an integer $m\ge 0$ such that $z{f}^{m}=0\text{;}$ hence $z/{f}^{n}=0$ in ${M}_{f},$ hence ${u}_{f}$ is injective. $\square $ 
Corollary (9.4). $\Gamma $ is exact on quasicoherent Modules over an affine scheme.
Proof.  
Let $\mathcal{F}\to \mathcal{G}\to \mathscr{H}$ be an exact sequence of quasicoherent $\stackrel{\sim}{A}\text{Modules.}$ By (9.2) and (9.3) this sequence is of the form $\stackrel{\sim}{M}\stackrel{\stackrel{\sim}{u}}{\to}N\stackrel{\stackrel{\sim}{v}}{\to}\stackrel{\sim}{P}$ $\text{(}M=\mathcal{F}\left(X\right),$ etc.). If $Q=\text{Im}\left(u\right),$ $R=\text{Ker}\left(v\right)$ then $\stackrel{\sim}{Q}=\stackrel{\sim}{R}$ (since the functor $M\mapsto \stackrel{\sim}{M}$ is exact), hence $Q=\stackrel{\sim}{Q}\left(X\right)=\stackrel{\sim}{R}\left(X\right)=R\text{.}$ Hence the sequence $M\to N\to P$ is exact, i.e. the sequence $\Gamma (X,\mathcal{F})\to \Gamma (X,\mathcal{G})\to \Gamma (X,\mathscr{H})$ is exact. $\square $ 
Theorem (9.5). Let $A$ be a Noetherian ring, $\mathcal{F}$ an $\stackrel{\sim}{A}\text{Module.}$ Then the following are equivalent:
(i)  $\mathcal{F}$ is coherent; 
(ii)  $\mathcal{F}$ is of finite type and quasicoherent; 
(iii)  $\mathcal{F}\cong \stackrel{\sim}{M}$ for some finitelygenerated $A\text{module}$ $M\text{.}$ 
Proof.  
(i) $\Rightarrow $ (ii) is always true (from the definitions). (ii) $\Rightarrow $ (iii): By (9.3) we have $\mathcal{F}\cong \stackrel{\sim}{M}$ for some $A\text{module}$ $M\text{.}$ Since $\mathcal{F}$ is of finite type and $X$ is quasicompact, there exists a finite covering of $X$ by basic open sets $D\left({f}_{i}\right),$ and exact sequences ${\stackrel{\sim}{A}}^{{p}_{i}}\to \mathcal{F}\to 0$ (over $D\left({f}_{i}\right)\text{),}$ i.e. exact sequences ${\stackrel{\sim}{A}}_{{f}_{i}}^{{p}_{i}}\to {\stackrel{\sim}{M}}_{{f}_{i}}\to 0\text{;}$ hence, by (9.4), exact sequences ${A}_{{f}_{i}}^{{p}_{i}}\to {M}_{{f}_{i}}\to 0\text{.}$ Thus each ${M}_{{f}_{i}}$ is a finitelygenerated ${A}_{{f}_{i}}\text{module,}$ generated say by ${t}_{ij}/1$ $(1\le j\le {p}_{i},{t}_{ij}\in M)\text{.}$ Let $N$ be the submodule of $M$ generated by all the ${t}_{ij}\text{.}$ If $z\in M,$ then $z/1\in {M}_{{f}_{i}}$ is of the form $\sum _{j}({t}_{ij}/1)\xb7({a}_{ij}/{f}_{i}^{{m}_{j}}),$ hence $z{f}_{i}^{m}\in N$ for all indices $i$ and some integer $m>0\text{.}$ Since the $D\left({f}_{i}\right)$ cover $X,$ the ${f}_{i}^{m}$ generate the unit ideal, i.e. we have an equation of the form $\sum _{i}{g}_{i}{f}_{i}^{m}=1,$ where ${g}_{i}\in A\text{.}$ Hence $z=\sum _{i}z{f}_{i}^{m}{g}_{i}\in N,$ consequently $M=N$ and therefore $M$ is finitely generated. (iii) $\Rightarrow $ (i). Suppose $\mathcal{F}=\stackrel{\sim}{M}$ where $M$ is a finitely generated $A\text{module.}$ Then we have an exact sequence of the form ${A}^{p}\to M\to 0$ for some integer $p\ge 0,$ hence $\stackrel{\sim}{A}\to \stackrel{\sim}{M}\to 0\text{;}$ thus $\stackrel{\sim}{M}$ is of finite type. It remains to show that if ${\stackrel{\sim}{A}}^{p}\to \stackrel{\sim}{M}$ over some open set (which we may take to be $D\left(f\right)$ for some $f\in A\text{),}$ then the kernel is of finite type. We have a homomorphism ${\stackrel{\sim}{A}}_{f}^{p}\to {\stackrel{\sim}{M}}_{f},$ hence a homomorphism ${A}_{f}^{p}\to {M}_{f}$ by (9.2); now ${A}_{f}$ is Noetherian (since $A$ is), hence the kernel is finitely generated. This completes the proof. Remark. The Noetherian assumption intervenes only in the proof of (iii) $\Rightarrow $ (i). $\square $ 
Corollary (9.6). If $A$ is Noetherian, $\stackrel{\sim}{A}$ is a coherent sheaf of rings.
Proposition (9.7). Let $\mathcal{F}$ be a quasicoherent $\stackrel{\sim}{A}\text{Module,}$ and let $\U0001d4ca$ be a covering of $X=\text{Spec}\left(A\right)$ by basic open sets $D\left({f}_{i}\right)\text{.}$ Then ${H}^{p}(\U0001d4ca,\mathcal{F})=0$ for all $p>0$ (and of course ${H}^{0}(\U0001d4ca,\mathcal{F})=\mathcal{F}\left(X\right)\text{).}$
Proof.  
By (9.3) we have $\mathcal{F}=\stackrel{\sim}{M},$ where $M=\mathcal{F}\left(X\right)$ is an $A\text{module.}$ Consider the $\u010c\text{ech}$ resolution of $\stackrel{\sim}{M}$ (Chapter 8): $$0\u27f6\stackrel{\sim}{M}\u27f6{C}^{0}(\U0001d4ca,\stackrel{\sim}{M})\u27f6{C}^{\prime}(\U0001d4ca,\stackrel{\sim}{M})\u27f6\dots $$whose sections over $X$ form the $\u010c\text{ech}$ complex $0\to M\to {C}^{0}(\U0001d4ca,\stackrel{\sim}{M})\to {C}^{1}(\U0001d4ca,\stackrel{\sim}{M})\to \dots \text{.}$ Recall that ${C}^{q}(\U0001d4ca,\stackrel{\sim}{M})$ is the sheaf associated with the pre sheaf $$D\left(g\right)\u27fc\prod _{\sigma}\stackrel{\sim}{M}({U}_{\sigma}\cap D\left(g\right))\text{;}$$now if $\sigma =({i}_{0},\dots ,{i}_{q})$ we have ${U}_{\sigma}\cap D\left(g\right)=D\left({f}_{{i}_{0}}\right)\cap \dots \cap D\left({f}_{{i}_{q}}\right)\cap D\left(g\right)=D\left({f}_{{i}_{0}}\dots {f}_{{i}_{q}}g\right)=D\left({f}_{\sigma}g\right)$ say; hence ${C}^{q}(\U0001d4ca,\stackrel{\sim}{M})$ is the sheaf associated with the presheaf $D\left(g\right)\mapsto \prod _{\sigma}{M}_{{f}_{\sigma}g}={\left(\prod _{\sigma}{M}_{{f}_{\sigma}}\right)}_{g},$ so that ${C}^{q}(\U0001d4ca,\stackrel{\sim}{M})={\left(\prod _{\sigma}{M}_{{f}_{\sigma}}\right)}^{\sim}\text{.}$ Hence, by (9.3), the sheaf ${C}^{q}(\U0001d4ca,\stackrel{\sim}{M})$ is quasicoherent; now $\Gamma $ is exact on quasicoherent sheaves (9.4), hence the $\u010c\text{ech}$ complex is exact, i.e. ${H}^{p}(\U0001d4ca,\mathcal{F})=0$ for all $p>0\text{.}$ $\square $ 
Theorem (9.8). If $X$ is an affine scheme and $\mathcal{F}$ a quasicoherent sheaf on $X,$ then ${H}^{p}(X,\mathcal{F})=0$ for all $p>0\text{.}$
Proof.  
Since finite basic open coverings are cofinal in the class of all open coverings of $X,$ it follows from (9.7) that ${\stackrel{\u02c7}{H}}^{p}(X,\mathcal{F})=0$ for all $p>0\text{.}$ Hence for any basic open set $U=D\left(f\right)$ we have ${\stackrel{\u02c7}{H}}^{q}(U,\mathcal{F}U)=0$ for all $q>0$ (since $U$ is an affine open set and $\mathcal{F}U$ is quasicoherent). Hence by Cartan's criterion (8.3) we have ${\stackrel{\u02c7}{H}}^{p}(X,\mathcal{F})={H}^{p}(X,\mathcal{F})$ for all $p>0\text{.}$ Hence ${H}^{p}(X,\mathcal{F})=0$ for all $p>0\text{.}$ $\square $ 
Remark. There is another proof, due to Chevalley, of (9.8) avoiding the use of (8.3) (which we didn't prove). Let $0\to \mathcal{F}\to {\mathcal{I}}^{0}\to {\mathcal{I}}^{1}\to \dots $ be an injective resolution of a quasicoherent sheaf $\mathcal{F}$ on $X\text{.}$ Then we have short exact sequences
$$\left({E}^{p}\right):\phantom{\rule{1em}{0ex}}0\u27f6{\mathcal{G}}^{p}\u27f6{\mathcal{I}}^{p}\u27f6{\mathcal{G}}^{p+1}\u27f60$$where ${\mathcal{G}}^{0}=\mathcal{F}$ and ${\mathcal{G}}^{p}=\text{Im}({\mathcal{I}}^{p1}\to {\mathcal{I}}^{p})$ for $p>0\text{.}$
Lemma (9.9). Let $f\in A$ and let $\U0001d4ca$ be any finite covering of $D\left(f\right)$ by basic open sets. Then ${H}^{q}(\U0001d4ca,{\mathcal{G}}^{p}D\left(f\right))=0$ for all $p\ge 0$ and all $q>0\text{.}$
Proof.  
Proof by induction on $p\text{.}$ True for $p=0$ by (9.7). Let $p\ge 0$ and assume (9.9) true for this value of $p$ (and all $q>0\text{).}$ Then ${H}^{1}(\U0001d4ca,{\mathcal{G}}^{p}D\left(f\right))=0$ for any finite covering of $D\left(f\right)$ by basic open sets. Since such open coverings of $D\left(f\right)$ are cofinal in the class of all open coverings of $D\left(f\right)$ it follows that ${\stackrel{\u02c7}{H}}^{1}(D\left(f\right),{\mathcal{G}}^{p}D\left(f\right))=0$ and therefore that ${H}^{1}(D\left(f\right),{\mathcal{G}}^{p}D\left(f\right))=0$ (since ${H}^{1}={\stackrel{\u02c7}{H}}^{1}$ always). Hence, from the exact cohomology sequence of $\left({E}^{p}\right),$ we have an exact sequence $$0\u27f6{\mathcal{G}}^{p}\left(D\left(f\right)\right)\u27f6{\mathcal{I}}^{p}\left(D\left(f\right)\right)\u27f6{\mathcal{G}}^{p+1}\left(D\left(f\right)\right)\u27f60\text{.}$$Since this sequence is exact for. every $f\in A,$ it follows that the sequence of $\u010c\text{ech}$ complexes $$\begin{array}{cc}\left(\u2736\right)& 0\u27f6{C}^{\u2022}(\U0001d4ca,{\mathcal{F}}^{p}D\left(f\right))\u27f6{C}^{\u2022}(\U0001d4ca,{\mathcal{I}}^{p}D\left(f\right))\u27f6C(\U0001d4ca,{\mathcal{G}}^{p+1}D\left(f\right))\u27f60\end{array}$$is exact. Now ${\mathcal{I}}^{p}$ is injective, hence its restriction to the open set $D\left(f\right)$ is injective and therefore the complex ${C}^{\u2022}(\U0001d4ca,{\mathcal{I}}^{p}D\left(f\right))$ is acyclic; consequently, from the cohomology exact sequence of $\left(\u2736\right),$ we get $${H}^{q}(\U0001d4ca,{\mathcal{G}}^{p+1}D\left(f\right))\cong {H}^{q+1}(\U0001d4ca,{\mathcal{G}}^{p}D\left(f\right))\phantom{\rule{2em}{0ex}}(q>0)$$and the term on the right is zero by the inductive hypothesis. $\square $ 
Taking $f=1,$ $q=1$ in (9.9), we have ${H}^{1}(\U0001d4ca,{\mathcal{G}}^{p})=0$ for all $p\ge 0,$ hence ${\stackrel{\u02c7}{H}}^{1}(X,{\mathcal{G}}^{p})=0,$ hence ${H}^{1}(X,{\mathcal{G}}^{p})=0\text{.}$ But from the exact sequences $\left({E}^{p}\right)$ we get (since each ${\mathcal{I}}^{p}$ is injective)
$${H}^{p}(X,\mathcal{F})={H}^{p}(X,{\mathcal{G}}^{0})\cong {H}^{p1}(X,{\mathcal{G}}^{1})\cong \dots \cong {H}^{1}(X,{\mathcal{G}}^{p1})=0\phantom{\rule{2em}{0ex}}(p>0)\text{.}$$Theorem (9.10). If $(X,{\mathcal{O}}_{X})$ is a scheme and $\mathcal{F}$ is a quasicoherent ${\mathcal{O}}_{X}\text{Module,}$ then ${H}^{q}(X,\mathcal{F})\cong {H}^{q}(\U0001d4ca,\mathcal{F})$ for any covering $\U0001d4ca$ of $X$ by affine open sets.
Proof.  
Let $\U0001d4ca={\left({U}_{i}\right)}_{i\in I}$ be an affine open covering of $X\text{.}$ Since, $X$ is a scheme, each ${U}_{\sigma}={U}_{{i}_{0}}\cap \dots \cap {U}_{{i}_{q}}$ is affine and hence by (9.8) ${H}^{p}({U}_{\sigma},\mathcal{F}{U}_{\sigma})=0$ for all $\sigma $ and all $p>0\text{.}$ Hence by the comparison theorem (8.2) we have ${H}^{p}(X,\mathcal{F})\cong {H}^{p}(\U0001d4ca,\mathcal{F})$ for all $p\ge 0\text{.}$ $\square $ 
Corollary (9.11). ${H}^{p}(X,\mathcal{F})\cong {\stackrel{\u02c7}{H}}^{p}(X,\mathcal{F})$ under the hypotheses of (9.10).
There is a converse of (9.8):
Theorem (9.12). (Serre's criterion.) Let $X$ be either a quasicompact scheme or a prescheme whose underlying space is Noetherian. If ${H}^{1}(X,\mathcal{F})=0$ for every quasicoherent ${\mathcal{O}}_{X}\text{Module}$ $\mathcal{F}$ (or even only for every quasicoherent Ideal $\mathcal{F}$ of ${\mathcal{O}}_{X}\text{),}$ then $X$ is an affine scheme.
For the proof we refer to (E.G.A., II, 5.2.1). (9.8) and (9.12) show that the vanishing of the ${H}^{p}(X,\mathcal{F})$ for $p>0$ and $\mathcal{F}$ quasicoherent characterizes affine schemes.
Let $X$ be a projective algebraic variety over an algebraically closed field $k,$ and let $\mathcal{F}$ be a coherent ${\mathcal{O}}_{X}\text{Module,}$ where ${\mathcal{O}}_{X}$ is the sheaf of local rings on $X\text{.}$ Serre proved that
(i)  ${H}^{q}(X,\mathcal{F})=0$ for $q>\text{dim}\hspace{0.17em}X\text{;}$ 
(ii)  ${H}^{q}(X,\mathcal{F})$ is a finitedimensional $k\text{vector}$ space for $0\le q\le \text{dim}\hspace{0.17em}X\text{.}$ 
The proof of (i) is easy: by (9.10) (or rather its counterpart for algebraic varieties) it is enough to find a covering of $X$ by $d+1$ affine open sets, where $d=\text{dim}\hspace{0.17em}X,$ and this can be achieved by intersecting $X$ by suitably chosen hyperplanes in the projective space $P$ in which $X$ is embedded. (ii) is proved by reducing to the case where $X=P$ and calculating the ${H}^{q}(P,\mathcal{F})$ quite explicitly.
Grothendieck subsequently generalized this theorem, firstly to the case where $X$ is complete (but not necessarily projective) and then to a statement about proper morphisms. If $f:X\to Y$ is a morphism of algebraic varieties, then ${f}_{\u2736}$ (Chapter 7) is a leftexact functor from ${\mathcal{O}}_{X}\text{Modules}$ to ${\mathcal{O}}_{Y}\text{Modules,}$ hence has right derived functors ${R}^{p}{f}_{\u2736}(p\ge 0)\text{.}$ Explicitly, if $\mathcal{F}$ is an ${\mathcal{O}}_{X}\text{Module,}$ ${R}^{p}{f}_{\u2736}\left(\mathcal{F}\right)$ is the sheaf on $Y$ associated to the presheaf $U\mapsto {H}^{p}({f}^{1}\left(U\right),\mathcal{F})$ $\text{(}U$ open in $Y\text{).}$ Then:
If $X,Y$ are algebraic varieties over $k,f:X\to Y$ a proper morphism, $\mathcal{F}$ a coherent ${\mathcal{O}}_{X}\text{Module,}$ then the 'higher direct images' ${R}^{p}{f}_{\u2736}\left(\mathcal{F}\right)$ are coherent ${\mathcal{O}}_{Y}\text{Modules.}$ (The statement for a complete variety $X$ is obtained by taking $Y$ to consist of a single point.)
Finally, this theorem generalizes to the case of a proper morphism of preschemes:
Let $X,Y$ be preschemes, $Y$ locally Noetherian (this means that $Y$ can be covered by affine open sets each of which is the scheme of a Noetherian ring). If $f:X\to Y$ is a proper morphism and $\mathcal{F}$ a coherent ${\mathcal{O}}_{X}\text{Module,}$ then the ${R}^{p}{f}_{\u2736}\left(\mathcal{F}\right)$ are coherent ${\mathcal{O}}_{Y}\text{Modules}$ (E.G.A., III, 3.2.1).
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes  I.G. Macdonald".