## Chapter 9

Last update: 11 June 2013

## Cohomology of Affine schemes

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 $ℬ={D\left(f\right)}_{f\in A}$ 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 $\Gamma \left(D\left(f\right),\stackrel{\sim }{M}\right)={M}_{f}$ for all $f\in A\text{.}$ In particular $\Gamma \left(X,\stackrel{\sim }{M}\right)=M\text{.}$

 Proof. Copy of the proof of (5.1) (ii). $\square$

Since formation of modules of fractions preserves exactness, it follows that the functor $M↦\stackrel{\sim }{M}$ (from $A\text{-modules}$ to $\stackrel{\sim }{A}\text{-Modules)}$ is exact. Moreover,

Corollary (9.2). If $M,N$ are $A\text{-modules,}$ then ${\text{Hom}}_{A}\left(M,N\right)\cong {\text{Hom}}_{\stackrel{\sim }{A}}\left(\stackrel{\sim }{M},\stackrel{\sim }{N}\right)\text{.}$

 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 ${\text{Hom}}_{A}\left(M,N\right)\to {\text{Hom}}_{\stackrel{\sim }{A}}\left(\stackrel{\sim }{M},\stackrel{\sim }{N}\right)\text{.}$ 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 ${\text{Hom}}_{\stackrel{\sim }{A}}\left(\stackrel{\sim }{M},\stackrel{\sim }{N}\right)\to {\text{Hom}}_{A}\left(M,N\right)\text{.}$ Verify that the two maps so defined are inverses of each other. $\square$

Theorem (9.3). Let $ℱ$ be an $A\text{-Module.}$ Then the following are equivalent:

(a) $ℱ\cong \stackrel{\sim }{M}$ 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 $ℱ|D\left({f}_{i}\right)\cong {\stackrel{\sim }{M}}_{i}$ for some ${A}_{{f}_{i}}\text{-module}$ ${M}_{i}$ and each index $i\text{;}$
(c) $ℱ$ is quasi-coherent;
(d) $ℱ$ satisfies the following two conditions:
 (${d}_{1}$) for each $g\in A$ and each $s\in ℱ\left(D\left(g\right)\right)$ there exists an integer $n\ge 0$ such that ${g}^{n}s$ can be extended to a global section of $ℱ$ (i.e. an element of $ℱ\left(X\right)\text{);}$ (${d}_{2}$) for each $g\in A$ and each $t\in ℱ\left(X\right)$ such that $t|D\left(g\right)=0,$ there exists $n\ge 0$ such that ${g}^{n}t=0\text{.}$

 Proof according to the scheme $(a) ⟹ (b) ⟺ (c). ⟺ ⟸ (d)$ (a) $⇒$ (b). Take the covering of $X$ consisting of the single set $D\left(1\right)=X\text{.}$ (b) $⇒$ (c). Since quasi-coherence is a local property, it is enough to prove (a) $⇒$ (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 $M↦\stackrel{\sim }{M}$ 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 quasi-coherent. (c) $⇒$ (b). Each $x\in X$ has a neighbourhood $D\left(f\right)$ over which $ℱ|D\left(f\right)$ 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) $ℱ|D\left(f\right)=\stackrel{\sim }{N}$ where $N=\text{coker}\left({A}_{f}^{\left(I\right)}\to {A}_{f}^{\left(J\right)}\right)\text{.}$ Since $X$ is quasi-compact, (b) is proved. (a) $⇒$ (d). If $g\in A$ and $s\in ℱ\left(D\left(g\right)\right)={M}_{g},$ 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 $M=ℱ\left(X\right)\text{.}$ 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) $⇒$ (d). We have to show that if each $ℱ|D\left({f}_{i}\right)$ satisfies (d), then so does $ℱ\text{.}$ Take $\left({d}_{2}\right)$ first. We have then $g\in A,$ $t\in ℱ\left(X\right)$ and $t|D\left(g\right)=0\text{.}$ Then $t|D\left(g{f}_{i}\right)=0$ (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 $ℱ|D\left({f}_{i}\right)$ there exists an integer ${n}_{i}\ge 0$ such that ${\left({f}_{i}g\right)}^{{n}_{i}}t|D\left({f}_{i}\right)=0,$ 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 $ℱ|D\left({f}_{i}\right),$ hence ${g}^{n}t$ is the zero section of $ℱ\text{.}$ To prove $\left({d}_{1}\right)\text{:}$ take $g\in A$ and $s\in ℱ\left(D\left(g\right)\right)\text{.}$ By applying $\left({d}_{1}\right)$ to $ℱ|D\left({f}_{i}\right),$ there exists an integer ${n}_{i}\ge 0$ and an element ${s}_{i}^{\prime }\in ℱ\left(D\left({f}_{i}\right)\right)$ which extends ${\left({f}_{i}g\right)}^{{n}_{i}}s|D\left({f}_{i}g\right)\text{.}$ 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 ${g}^{{n}_{i}}s|D\left({f}_{i}g\right)\text{;}$ 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 $ℱ|D\left({f}_{i}\right)={\stackrel{\sim }{M}}_{i},$ it follows that each $ℱ|D\left({f}_{i}\right)\cap D\left({f}_{j}\right)$ satisfies (a) and therefore (d), hence by $\left({d}_{2}\right)$ applied to $ℱ|D\left({f}_{i}\right)\cap D\left({f}_{j}\right)$ there exist integers ${m}_{ij}\ge 0$ such that ${\left({f}_{i}{f}_{j}g\right)}^{{m}_{ij}}\left({s}_{i}-{s}_{j}\right)$ 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 $ℱ\left(D\left({f}_{i}{f}_{j}\right)\right),$ hence ${g}^{m}\left({s}_{i}-{s}_{j}\right)$ 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 ${g}^{m}{s}_{i}\in \Gamma \left(D\left({f}_{i}\right),ℱ\right)$ are all of them restrictions of a global section $s\prime$ of $ℱ\text{.}$ This section $s\prime$ is an extension of ${g}^{m+n}s,$ hence $\left({d}_{1}\right)$ is proved. (d) $⇒$ (a). Let $M=ℱ\left(X\right)=\Gamma \left(X,ℱ\right)\text{.}$ We shall define a homomorphism $u:\stackrel{\sim }{M}\to ℱ$ and show that it is an isomorphism. For this we must define ${u}_{f}:{M}_{f}\to ℱ\left(D\left(f\right)\right)$ for each $f\in A,$ satisfying the usual compatibility conditions. Start with the restriction homomorphism $ℱ\left(X\right)\to ℱ\left(D\left(f\right)\right),$ i.e. $M\to ℱ\left(D\left(f\right)\right)\text{.}$ Since $f$ is a unit in ${A}_{f},$ this homomorphism factorizes through ${M}_{f}\text{:}$ $M⟶{M}_{f}\stackrel{{u}_{f}}{⟶}ℱ\left(D\left(f\right)\right)\text{.}$ 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 $ℱ\left(D\left(f\right)\right)\text{.}$ Then by $\left({d}_{1}\right)$ ${f}^{n}s$ lifts to a global section of $ℱ,$ 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 ${u}_{f}\left(z/{f}^{n}\right)=0,$ then ${u}_{f}\left(z/1\right)=0$ and therefore the restriction of $z \left(\in ℱ\left(X\right)\right)$ 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 quasi-coherent Modules over an affine scheme.

 Proof. Let $ℱ\to 𝒢\to ℋ$ be an exact sequence of quasi-coherent $\stackrel{\sim }{A}\text{-Modules.}$ By (9.2) and (9.3) this sequence is of the form $\stackrel{\sim }{M}\stackrel{\stackrel{\sim }{u}}{\to }N\stackrel{\stackrel{\sim }{v}}{\to }\stackrel{\sim }{P}$ $\text{(}M=ℱ\left(X\right),$ etc.). If $Q=\text{Im}\left(u\right),$ $R=\text{Ker}\left(v\right)$ then $\stackrel{\sim }{Q}=\stackrel{\sim }{R}$ (since the functor $M↦\stackrel{\sim }{M}$ 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 $\Gamma \left(X,ℱ\right)\to \Gamma \left(X,𝒢\right)\to \Gamma \left(X,ℋ\right)$ is exact. $\square$

Theorem (9.5). Let $A$ be a Noetherian ring, $ℱ$ an $\stackrel{\sim }{A}\text{-Module.}$ Then the following are equivalent:

 (i) $ℱ$ is coherent; (ii) $ℱ$ is of finite type and quasi-coherent; (iii) $ℱ\cong \stackrel{\sim }{M}$ for some finitely-generated $A\text{-module}$ $M\text{.}$

 Proof. (i) $⇒$ (ii) is always true (from the definitions). (ii) $⇒$ (iii): By (9.3) we have $ℱ\cong \stackrel{\sim }{M}$ for some $A\text{-module}$ $M\text{.}$ Since $ℱ$ is of finite type and $X$ is quasi-compact, there exists a finite covering of $X$ by basic open sets $D\left({f}_{i}\right),$ and exact sequences ${\stackrel{\sim }{A}}^{{p}_{i}}\to ℱ\to 0$ (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 finitely-generated ${A}_{{f}_{i}}\text{-module,}$ generated say by ${t}_{ij}/1$ $\left(1\le j\le {p}_{i},{t}_{ij}\in M\right)\text{.}$ 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 $\sum _{j}\left({t}_{ij}/1\right)·\left({a}_{ij}/{f}_{i}^{{m}_{j}}\right),$ 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) $⇒$ (i). Suppose $ℱ=\stackrel{\sim }{M}$ 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) $⇒$ (i). $\square$

Corollary (9.6). If $A$ is Noetherian, $\stackrel{\sim }{A}$ is a coherent sheaf of rings.

Proposition (9.7). Let $ℱ$ be a quasi-coherent $\stackrel{\sim }{A}\text{-Module,}$ and let $𝓊$ be a covering of $X=\text{Spec}\left(A\right)$ by basic open sets $D\left({f}_{i}\right)\text{.}$ Then ${H}^{p}\left(𝓊,ℱ\right)=0$ for all $p>0$ (and of course ${H}^{0}\left(𝓊,ℱ\right)=ℱ\left(X\right)\text{).}$

 Proof. By (9.3) we have $ℱ=\stackrel{\sim }{M},$ where $M=ℱ\left(X\right)$ is an $A\text{-module.}$ Consider the $Č\text{ech}$ resolution of $\stackrel{\sim }{M}$ (Chapter 8): $0⟶M∼⟶C0 (𝓊,M∼)⟶ C′(𝓊,M∼) ⟶…$ whose sections over $X$ form the $Č\text{ech}$ complex $0\to M\to {C}^{0}\left(𝓊,\stackrel{\sim }{M}\right)\to {C}^{1}\left(𝓊,\stackrel{\sim }{M}\right)\to \dots \text{.}$ Recall that ${C}^{q}\left(𝓊,\stackrel{\sim }{M}\right)$ is the sheaf associated with the pre sheaf $D(g)⟼∏σM∼ (Uσ∩D(g));$ now if $\sigma =\left({i}_{0},\dots ,{i}_{q}\right)$ 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 ${C}^{q}\left(𝓊,\stackrel{\sim }{M}\right)$ is the sheaf associated with the presheaf $D\left(g\right)↦\prod _{\sigma }{M}_{{f}_{\sigma }g}={\left(\prod _{\sigma }{M}_{{f}_{\sigma }}\right)}_{g},$ so that ${C}^{q}\left(𝓊,\stackrel{\sim }{M}\right)={\left(\prod _{\sigma }{M}_{{f}_{\sigma }}\right)}^{\sim }\text{.}$ Hence, by (9.3), the sheaf ${C}^{q}\left(𝓊,\stackrel{\sim }{M}\right)$ is quasi-coherent; now $\Gamma$ is exact on quasi-coherent sheaves (9.4), hence the $Č\text{ech}$ complex is exact, i.e. ${H}^{p}\left(𝓊,ℱ\right)=0$ for all $p>0\text{.}$ $\square$

Theorem (9.8). If $X$ is an affine scheme and $ℱ$ a quasi-coherent sheaf on $X,$ then ${H}^{p}\left(X,ℱ\right)=0$ 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 ${\stackrel{ˇ}{H}}^{p}\left(X,ℱ\right)=0$ for all $p>0\text{.}$ Hence for any basic open set $U=D\left(f\right)$ we have ${\stackrel{ˇ}{H}}^{q}\left(U,ℱ|U\right)=0$ for all $q>0$ (since $U$ is an affine open set and $ℱ|U$ is quasi-coherent). Hence by Cartan's criterion (8.3) we have ${\stackrel{ˇ}{H}}^{p}\left(X,ℱ\right)={H}^{p}\left(X,ℱ\right)$ for all $p>0\text{.}$ Hence ${H}^{p}\left(X,ℱ\right)=0$ 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 $0\to ℱ\to {ℐ}^{0}\to {ℐ}^{1}\to \dots$ be an injective resolution of a quasi-coherent sheaf $ℱ$ on $X\text{.}$ Then we have short exact sequences

$(Ep):0⟶ 𝒢p⟶ℐp⟶ 𝒢p+1⟶0$

where ${𝒢}^{0}=ℱ$ and ${𝒢}^{p}=\text{Im}\left({ℐ}^{p-1}\to {ℐ}^{p}\right)$ for $p>0\text{.}$

Lemma (9.9). Let $f\in A$ and let $𝓊$ be any finite covering of $D\left(f\right)$ by basic open sets. Then ${H}^{q}\left(𝓊,{𝒢}^{p}|D\left(f\right)\right)=0$ 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 ${H}^{1}\left(𝓊,{𝒢}^{p}|D\left(f\right)\right)=0$ 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 ${\stackrel{ˇ}{H}}^{1}\left(D\left(f\right),{𝒢}^{p}|D\left(f\right)\right)=0$ and therefore that ${H}^{1}\left(D\left(f\right),{𝒢}^{p}|D\left(f\right)\right)=0$ (since ${H}^{1}={\stackrel{ˇ}{H}}^{1}$ always). Hence, from the exact cohomology sequence of $\left({E}^{p}\right),$ we have an exact sequence $0⟶𝒢p(D(f))⟶ ℐp(D(f))⟶ 𝒢p+1(D(f)) ⟶0.$ Since this sequence is exact for. every $f\in A,$ it follows that the sequence of $Č\text{ech}$ complexes $(✶) 0⟶C• (𝓊,ℱp|D(f)) ⟶C• (𝓊,ℐp|D(f)) ⟶C(𝓊,𝒢p+1|D(f)) ⟶0$ is exact. Now ${ℐ}^{p}$ is injective, hence its restriction to the open set $D\left(f\right)$ is injective and therefore the complex ${C}^{•}\left(𝓊,{ℐ}^{p}|D\left(f\right)\right)$ is acyclic; consequently, from the cohomology exact sequence of $\left(✶\right),$ we get $Hq(𝓊,𝒢p+1|D(f)) ≅Hq+1 (𝓊,𝒢p|D(f)) (q>0)$ and the term on the right is zero by the inductive hypothesis. $\square$

Taking $f=1,$ $q=1$ in (9.9), we have ${H}^{1}\left(𝓊,{𝒢}^{p}\right)=0$ for all $p\ge 0,$ hence ${\stackrel{ˇ}{H}}^{1}\left(X,{𝒢}^{p}\right)=0,$ hence ${H}^{1}\left(X,{𝒢}^{p}\right)=0\text{.}$ But from the exact sequences $\left({E}^{p}\right)$ we get (since each ${ℐ}^{p}$ is injective)

$Hp(X,ℱ)= Hp(X,𝒢0) ≅Hp-1 (X,𝒢1)≅…≅ H1(X,𝒢p-1) =0(p>0).$

Theorem (9.10). If $\left(X,{𝒪}_{X}\right)$ is a scheme and $ℱ$ is a quasi-coherent ${𝒪}_{X}\text{-Module,}$ then ${H}^{q}\left(X,ℱ\right)\cong {H}^{q}\left(𝓊,ℱ\right)$ for any covering $𝓊$ of $X$ by affine open sets.

 Proof. Let $𝓊={\left({U}_{i}\right)}_{i\in I}$ 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) ${H}^{p}\left({U}_{\sigma },ℱ|{U}_{\sigma }\right)=0$ for all $\sigma$ and all $p>0\text{.}$ Hence by the comparison theorem (8.2) we have ${H}^{p}\left(X,ℱ\right)\cong {H}^{p}\left(𝓊,ℱ\right)$ for all $p\ge 0\text{.}$ $\square$

Corollary (9.11). ${H}^{p}\left(X,ℱ\right)\cong {\stackrel{ˇ}{H}}^{p}\left(X,ℱ\right)$ under the hypotheses of (9.10).

There is a converse of (9.8):

Theorem (9.12). (Serre's criterion.) Let $X$ be either a quasi-compact scheme or a prescheme whose underlying space is Noetherian. If ${H}^{1}\left(X,ℱ\right)=0$ for every quasi-coherent ${𝒪}_{X}\text{-Module}$ $ℱ$ (or even only for every quasi-coherent Ideal $ℱ$ of ${𝒪}_{X}\text{),}$ 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 ${H}^{p}\left(X,ℱ\right)$ for $p>0$ and $ℱ$ quasi-coherent characterizes affine schemes.

Let $X$ be a projective algebraic variety over an algebraically closed field $k,$ and let $ℱ$ be a coherent ${𝒪}_{X}\text{-Module,}$ where ${𝒪}_{X}$ is the sheaf of local rings on $X\text{.}$ Serre proved that

 (i) ${H}^{q}\left(X,ℱ\right)=0$ for $q>\text{dim} X\text{;}$ (ii) ${H}^{q}\left(X,ℱ\right)$ is a finite-dimensional $k\text{-vector}$ space for $0\le q\le \text{dim} X\text{.}$

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 $d=\text{dim} X,$ 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 ${H}^{q}\left(P,ℱ\right)$ 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 ${f}_{✶}$ (Chapter 7) is a left-exact functor from ${𝒪}_{X}\text{-Modules}$ to ${𝒪}_{Y}\text{-Modules,}$ hence has right derived functors ${R}^{p}{f}_{✶}\left(p\ge 0\right)\text{.}$ Explicitly, if $ℱ$ is an ${𝒪}_{X}\text{-Module,}$ ${R}^{p}{f}_{✶}\left(ℱ\right)$ is the sheaf on $Y$ associated to the presheaf $U↦{H}^{p}\left({f}^{-1}\left(U\right),ℱ\right)$ $\text{(}U$ open in $Y\text{).}$ Then:

If $X,Y$ are algebraic varieties over $k,f:X\to Y$ a proper morphism, $ℱ$ a coherent ${𝒪}_{X}\text{-Module,}$ then the 'higher direct images' ${R}^{p}{f}_{✶}\left(ℱ\right)$ are coherent ${𝒪}_{Y}\text{-Modules.}$ (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 $ℱ$ a coherent ${𝒪}_{X}\text{-Module,}$ then the ${R}^{p}{f}_{✶}\left(ℱ\right)$ are coherent ${𝒪}_{Y}\text{-Modules}$ (E.G.A., III, 3.2.1).

## Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".