Chapter 9: Cohomology of Affine schemes

Chapter 9

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 11 June 2013

Cohomology of Affine schemes

Let $A$ be any commutative ring with identity element, and let $X = Spec (A) .$ We recall that, for any $f \in A,$ the 'basic open set' $D (f)$ is the set of all $x \in X$ such that $f (x) \neq 0,$ i.e. such that $f \notin j_{x} .$ Let $M$ be any $A -module;$ then we can form the module of fractions $M_{f},$ whose elements are all fractions of the form $m / f^{n}$ $(m \in M$ $n$ an integer $\geq 0).$ $M_{f}$ is a module over the ring $A_{f} .$ We may then consider the presheaf $D (f) \to M_{f},$ defined on the basis $ℬ = {D (f)}_{f \in A}$ of $X .$ We denote by $\tilde{M}$ the presheaf on $X$ which it determines. Since each $M_{f}$ is an $A_{f} -module,$ $\tilde{M}$ is an $\tilde{A} -Module.$

Proposition (9.1). $\tilde{M}$ is a sheaf, and hence $Γ (D (f), \tilde{M}) = M_{f}$ for all $f \in A .$ In particular $Γ (X, \tilde{M}) = M .$

Proof.

Copy of the proof of (5.1) (ii).

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

Corollary (9.2). If $M, N$ are $A -modules,$ then ${Hom}_{A} (M, N) ≅ {Hom}_{\tilde{A}} (\tilde{M}, \tilde{N}) .$

Proof.

$φ : M \to N$ gives rise to $φ_{f} : M_{f} \to N_{f}$ for each $f \in A,$ hence to $\tilde{φ} : \tilde{M} \to \tilde{N} .$ Hence we have a homomorphism ${Hom}_{A} (M, N) \to {Hom}_{\tilde{A}} (\tilde{M}, \tilde{N}) .$ Conversely, given $u : \tilde{M} \to \tilde{N}$ we have $u (X) : \tilde{M} (X) \to \tilde{N} (X),$ i.e. by (9.1) a homomorphism $M \to N .$ Hence a map ${Hom}_{\tilde{A}} (\tilde{M}, \tilde{N}) \to {Hom}_{A} (M, N) .$ Verify that the two maps so defined are inverses of each other.

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

(a)

$ℱ ≅ \tilde{M}$ for some $A -module$ $M;$

(b)

there exists a finite open covering of $X$ by basic open sets $D (f_{i})$ such that $ℱ | D (f_{i}) ≅ {\tilde{M}}_{i}$ for some $A_{f_{i}} -module$ $M_{i}$ and each index $i;$

(c)

$ℱ$ is quasi-coherent;

(d)

$ℱ$ satisfies the following two conditions:

( $d_{1}$ )	for each $g \in A$ and each $s \in ℱ (D (g))$ there exists an integer $n \geq 0$ such that $g^{n} s$ can be extended to a global section of $ℱ$ (i.e. an element of $ℱ (X));$
( $d_{2}$ )	for each $g \in A$ and each $t \in ℱ (X)$ such that $t \| D (g) = 0,$ there exists $n \geq 0$ such that $g^{n} t = 0 .$

Proof according to the scheme

\begin{matrix} (a) & ⟹ & (b) & ⟺ & (c). \\ (d) \end{matrix}

(a) $\Rightarrow$ (b). Take the covering of $X$ consisting of the single set $D (1) = X .$

(b) $\Rightarrow$ (c). Since quasi-coherence is a local property, it is enough to prove (a) $\Rightarrow$ (c). We have an exact sequence $A^{(I)} \to A^{(J)} \to M \to 0,$ where $A^{(I)}, A^{(J)}$ are direct sums of copies of $A;$ hence, since $M \mapsto \tilde{M}$ is an exact functor, an exact functor, an exact sequence ${\tilde{A}}^{(I)} \to {\tilde{A}}^{(J)} \to \tilde{M} \to 0 .$ Hence $\tilde{M}$ is quasi-coherent.

(c) $\Rightarrow$ (b). Each $x \in X$ has a neighbourhood $D (f)$ over which $ℱ | D (f)$ is the cokernel of a homomorphism ${\tilde{A}}^{(I)} | D (f) \to {\tilde{A}}^{(J)} | D (f),$ i.e. a homomorphism ${\tilde{A}}_{f}^{(I)} \to {\tilde{A}}_{f}^{(J)} .$ Hence by (9.2) $ℱ | D (f) = \tilde{N}$ where $N = coker (A_{f}^{(I)} \to A_{f}^{(J)}) .$ Since $X$ is quasi-compact, (b) is proved.

(a) $\Rightarrow$ (d). If $g \in A$ and $s \in ℱ (D (g)) = M_{g},$ then $s = m / g^{n}$ for some $m \in M$ and some integer $n \geq 0,$ hence $s g^{n} = m / 1 =$ image of $m$ in $M_{g},$ i.e. $s g^{n}$ is the image of an element of $M = ℱ (X) .$ If $g \in A,$ $t \in M$ and $t / 1 = 0$ in $M_{g},$ then $t g^{n} = 0$ for some integer $n \geq 0,$ from the basic properties of modules of fractions.

(b) $\Rightarrow$ (d). We have to show that if each $ℱ | D (f_{i})$ satisfies (d), then so does $ℱ .$ Take $(d_{2})$ first. We have then $g \in A,$ $t \in ℱ (X)$ and $t | D (g) = 0 .$ Then $t | D (g f_{i}) = 0$ (since $D (g f_{i}) = D (f) \cap D (f_{i}));$ hence by $(d_{2})$ applied to $ℱ | D (f_{i})$ there exists an integer $n_{i} \geq 0$ such that ${(f_{i} g)}^{n_{i}} t | D (f_{i}) = 0,$ i.e. ${(f_{i} g)}^{n_{i}} t = 0$ in $M_{i};$ now $f_{i}$ is a unit in $A_{f_{i}},$ hence $g^{n_{i}} t = 0$ in $M_{i} .$ Let $n$ be the largest of the $n_{i},$ then we have $g^{n} t = 0$ in each $ℱ | D (f_{i}),$ hence $g^{n} t$ is the zero section of $ℱ .$

To prove $(d_{1}) :$ take $g \in A$ and $s \in ℱ (D (g)) .$ By applying $(d_{1})$ to $ℱ | D (f_{i}),$ there exists an integer $n_{i} \geq 0$ and an element $s_{i}^{'} \in ℱ (D (f_{i}))$ which extends ${(f_{i} g)}^{n_{i}} s | D (f_{i} g) .$ Since $f_{i}$ is a unit in $A_{f_{i}},$ there exists $s_{i}^{'} = f_{i}^{n_{i}} s_{i},$ and $s_{i}$ extends $g^{n_{i}} s | D (f_{i} g);$ and we may take all the $n_{i}$ to be equal, say $n_{i} = n .$ By construction, $s_{i} - s_{j}$ restricted to $D (f_{i} f_{j} g)$ is zero; now since $ℱ | D (f_{i}) = {\tilde{M}}_{i},$ it follows that each $ℱ | D (f_{i}) \cap D (f_{j})$ satisfies (a) and therefore (d), hence by $(d_{2})$ applied to $ℱ | D (f_{i}) \cap D (f_{j})$ there exist integers $m_{i j} \geq 0$ such that ${(f_{i} f_{j} g)}^{m_{i j}} (s_{i} - s_{j})$ restricted to $D (f_{i}) \cap D (f_{j}) = D (f_{i} f_{j})$ is zero; but $f_{i} f_{j}$ is a unit in $ℱ (D (f_{i} f_{j})),$ hence $g^{m} (s_{i} - s_{j})$ restricted to $D (f_{i}) \cap D (f_{j})$ is zero, where $m = max (m_{i j}) .$ Hence the $g^{m} s_{i} \in Γ (D (f_{i}), ℱ)$ are all of them restrictions of a global section $s'$ of $ℱ .$ This section $s'$ is an extension of $g^{m + n} s,$ hence $(d_{1})$ is proved.

(d) $\Rightarrow$ (a). Let $M = ℱ (X) = Γ (X, ℱ) .$ We shall define a homomorphism $u : \tilde{M} \to ℱ$ and show that it is an isomorphism. For this we must define $u_{f} : M_{f} \to ℱ (D (f))$ for each $f \in A,$ satisfying the usual compatibility conditions. Start with the restriction homomorphism $ℱ (X) \to ℱ (D (f)),$ i.e. $M \to ℱ (D (f)) .$ Since $f$ is a unit in $A_{f},$ this homomorphism factorizes through $M_{f} :$ $M ⟶ M_{f} \overset{u_{f}}{⟶} ℱ (D (f)) .$ This defines $u_{f} .$ We shall show that $(d_{1})$ implies $u_{f}$ surjective, and $(d_{2})$ implies $u_{f}$ injective.

Let $s$ be any element of $ℱ (D (f)) .$ Then by $(d_{1})$ $f^{n} s$ lifts to a global section of $ℱ,$ for some integer $n \geq 0,$ i.e. $f^{n} s$ is in the image of $M,$ hence is in $u_{f} (M_{f}) .$ Hence, as $f$ is a unit in $A_{f},$ we have $s \in u_{f} (M_{f})$ 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 $z (\in ℱ (X))$ to $D (f)$ is zero; hence by $(d_{2})$ there exists an integer $m \geq 0$ such that $z f^{m} = 0;$ hence $z / f^{n} = 0$ in $M_{f},$ hence $u_{f}$ is injective.

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Corollary (9.4). $Γ$ is exact on quasi-coherent Modules over an affine scheme.

Proof.

Let $ℱ \to 𝒢 \to ℋ$ be an exact sequence of quasi-coherent $\tilde{A} -Modules.$ By (9.2) and (9.3) this sequence is of the form $\tilde{M} \overset{\tilde{u}}{\to} N \overset{\tilde{v}}{\to} \tilde{P}$ $(M = ℱ (X),$ etc.). If $Q = Im (u),$ $R = Ker (v)$ then $\tilde{Q} = \tilde{R}$ (since the functor $M \mapsto \tilde{M}$ is exact), hence $Q = \tilde{Q} (X) = \tilde{R} (X) = R .$ Hence the sequence $M \to N \to P$ is exact, i.e. the sequence $Γ (X, ℱ) \to Γ (X, 𝒢) \to Γ (X, ℋ)$ is exact.

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

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

Proof.

(i) $\Rightarrow$ (ii) is always true (from the definitions).

(ii) $\Rightarrow$ (iii): By (9.3) we have $ℱ ≅ \tilde{M}$ for some $A -module$ $M .$ Since $ℱ$ is of finite type and $X$ is quasi-compact, there exists a finite covering of $X$ by basic open sets $D (f_{i}),$ and exact sequences ${\tilde{A}}^{p_{i}} \to ℱ \to 0$ (over $D (f_{i})),$ i.e. exact sequences ${\tilde{A}}_{f_{i}}^{p_{i}} \to {\tilde{M}}_{f_{i}} \to 0;$ hence, by (9.4), exact sequences $A_{f_{i}}^{p_{i}} \to M_{f_{i}} \to 0 .$ Thus each $M_{f_{i}}$ is a finitely-generated $A_{f_{i}} -module,$ generated say by $t_{i j} / 1$ $(1 \leq j \leq p_{i}, t_{i j} \in M) .$ Let $N$ be the submodule of $M$ generated by all the $t_{i j} .$ If $z \in M,$ then $z / 1 \in M_{f_{i}}$ is of the form $\sum_{j} (t_{i j} / 1) \cdot (a_{i j} / f_{i}^{m_{j}}),$ hence $z f_{i}^{m} \in N$ for all indices $i$ and some integer $m > 0 .$ Since the $D (f_{i})$ 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 $g_{i} \in A .$ 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 $ℱ = \tilde{M}$ where $M$ is a finitely generated $A -module.$ Then we have an exact sequence of the form $A^{p} \to M \to 0$ for some integer $p \geq 0,$ hence $\tilde{A} \to \tilde{M} \to 0;$ thus $\tilde{M}$ is of finite type. It remains to show that if ${\tilde{A}}^{p} \to \tilde{M}$ over some open set (which we may take to be $D (f)$ for some $f \in A),$ then the kernel is of finite type. We have a homomorphism ${\tilde{A}}_{f}^{p} \to {\tilde{M}}_{f},$ 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).

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

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

Proof.

By (9.3) we have $ℱ = \tilde{M},$ where $M = ℱ (X)$ is an $A -module.$ Consider the $Č ech$ resolution of $\tilde{M}$ (Chapter 8):

0 ⟶ \tilde{M} ⟶ C^{0} (𝓊, \tilde{M}) ⟶ C^{'} (𝓊, \tilde{M}) ⟶ \dots

whose sections over $X$ form the $Č ech$ complex $0 \to M \to C^{0} (𝓊, \tilde{M}) \to C^{1} (𝓊, \tilde{M}) \to \dots .$ Recall that $C^{q} (𝓊, \tilde{M})$ is the sheaf associated with the pre sheaf

D (g) ⟼ \prod_{σ} \tilde{M} (U_{σ} \cap D (g));

now if $σ = (i_{0}, \dots, i_{q})$ we have $U_{σ} \cap D (g) = D (f_{i_{0}}) \cap \dots \cap D (f_{i_{q}}) \cap D (g) = D (f_{i_{0}} \dots f_{i_{q}} g) = D (f_{σ} g)$ say; hence $C^{q} (𝓊, \tilde{M})$ is the sheaf associated with the presheaf $D (g) \mapsto \prod_{σ} M_{f_{σ} g} = {(\prod_{σ} M_{f_{σ}})}_{g},$ so that $C^{q} (𝓊, \tilde{M}) = {(\prod_{σ} M_{f_{σ}})}^{\sim} .$ Hence, by (9.3), the sheaf $C^{q} (𝓊, \tilde{M})$ is quasi-coherent; now $Γ$ is exact on quasi-coherent sheaves (9.4), hence the $Č ech$ complex is exact, i.e. $H^{p} (𝓊, ℱ) = 0$ for all $p > 0 .$

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Theorem (9.8). If $X$ is an affine scheme and $ℱ$ a quasi-coherent sheaf on $X,$ then $H^{p} (X, ℱ) = 0$ for all $p > 0 .$

Proof.

Since finite basic open coverings are cofinal in the class of all open coverings of $X,$ it follows from (9.7) that ${\overset{ˇ}{H}}^{p} (X, ℱ) = 0$ for all $p > 0 .$ Hence for any basic open set $U = D (f)$ we have ${\overset{ˇ}{H}}^{q} (U, ℱ | U) = 0$ 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 ${\overset{ˇ}{H}}^{p} (X, ℱ) = H^{p} (X, ℱ)$ for all $p > 0 .$ Hence $H^{p} (X, ℱ) = 0$ for all $p > 0 .$

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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 $X .$ Then we have short exact sequences

(E^{p}) : 0 ⟶ 𝒢^{p} ⟶ ℐ^{p} ⟶ 𝒢^{p + 1} ⟶ 0

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

Lemma (9.9). Let $f \in A$ and let $𝓊$ be any finite covering of $D (f)$ by basic open sets. Then $H^{q} (𝓊, 𝒢^{p} | D (f)) = 0$ for all $p \geq 0$ and all $q > 0 .$

Proof.

Proof by induction on $p .$ True for $p = 0$ by (9.7). Let $p \geq 0$ and assume (9.9) true for this value of $p$ (and all $q > 0).$ Then $H^{1} (𝓊, 𝒢^{p} | D (f)) = 0$ for any finite covering of $D (f)$ by basic open sets. Since such open coverings of $D (f)$ are cofinal in the class of all open coverings of $D (f)$ it follows that ${\overset{ˇ}{H}}^{1} (D (f), 𝒢^{p} | D (f)) = 0$ and therefore that $H^{1} (D (f), 𝒢^{p} | D (f)) = 0$ (since $H^{1} = {\overset{ˇ}{H}}^{1}$ always). Hence, from the exact cohomology sequence of $(E^{p}),$ we have an exact sequence

0 ⟶ 𝒢^{p} (D (f)) ⟶ ℐ^{p} (D (f)) ⟶ 𝒢^{p + 1} (D (f)) ⟶ 0 .

Since this sequence is exact for. every $f \in A,$ it follows that the sequence of $Č ech$ complexes

\begin{matrix} (✶) & 0 ⟶ C^{•} (𝓊, ℱ^{p} | D (f)) ⟶ C^{•} (𝓊, ℐ^{p} | D (f)) ⟶ C (𝓊, 𝒢^{p + 1} | D (f)) ⟶ 0 \end{matrix}

is exact. Now $ℐ^{p}$ is injective, hence its restriction to the open set $D (f)$ is injective and therefore the complex $C^{•} (𝓊, ℐ^{p} | D (f))$ is acyclic; consequently, from the cohomology exact sequence of $(✶),$ we get

H^{q} (𝓊, 𝒢^{p + 1} | D (f)) ≅ H^{q + 1} (𝓊, 𝒢^{p} | D (f)) (q > 0)

and the term on the right is zero by the inductive hypothesis.

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

H^{p} (X, ℱ) = H^{p} (X, 𝒢^{0}) ≅ H^{p - 1} (X, 𝒢^{1}) ≅ \dots ≅ H^{1} (X, 𝒢^{p - 1}) = 0 (p > 0) .

Theorem (9.10). If $(X, 𝒪_{X})$ is a scheme and $ℱ$ is a quasi-coherent $𝒪_{X} -Module,$ then $H^{q} (X, ℱ) ≅ H^{q} (𝓊, ℱ)$ for any covering $𝓊$ of $X$ by affine open sets.

Proof.

Let $𝓊 = {(U_{i})}_{i \in I}$ be an affine open covering of $X .$ Since, $X$ is a scheme, each $U_{σ} = U_{i_{0}} \cap \dots \cap U_{i_{q}}$ is affine and hence by (9.8) $H^{p} (U_{σ}, ℱ | U_{σ}) = 0$ for all $σ$ and all $p > 0 .$ Hence by the comparison theorem (8.2) we have $H^{p} (X, ℱ) ≅ H^{p} (𝓊, ℱ)$ for all $p \geq 0 .$

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Corollary (9.11). $H^{p} (X, ℱ) ≅ {\overset{ˇ}{H}}^{p} (X, ℱ)$ under the hypotheses of (9.10).

There is a converse of (9.8):

Theorem (9.12). (Serre's criterion.) Let $X$ be either a quasi-compact scheme or a prescheme whose underlying space is Noetherian. If $H^{1} (X, ℱ) = 0$ for every quasi-coherent $𝒪_{X} -Module$ $ℱ$ (or even only for every quasi-coherent Ideal $ℱ$ of $𝒪_{X}),$ 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 $H^{p} (X, ℱ)$ 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 $𝒪_{X} -Module,$ where $𝒪_{X}$ is the sheaf of local rings on $X .$ Serre proved that

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

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

If $X, Y$ are algebraic varieties over $k, f : X \to Y$ a proper morphism, $ℱ$ a coherent $𝒪_{X} -Module,$ then the 'higher direct images' $R^{p} f_{✶} (ℱ)$ are coherent $𝒪_{Y} -Modules.$ (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 $𝒪_{X} -Module,$ then the $R^{p} f_{✶} (ℱ)$ are coherent $𝒪_{Y} -Modules$ (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".

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