The Stacks project

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29.3 Vanishing of cohomology

We have seen that on an affine scheme the higher cohomology groups of any quasi-coherent sheaf vanish (Lemma 29.2.2). It turns out that this also characterizes affine schemes. We give two versions.

slogan, EGA] '>reference

Lemma 29.3.1. Let $X$ be a scheme. Assume that

  1. $X$ is quasi-compact,

  2. for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ we have $H^1(X, \mathcal{I}) = 0$.

Then $X$ is affine.

Proof. Let $x \in X$ be a closed point. Let $U \subset X$ be an affine open neighbourhood of $x$. Write $U = \mathop{\mathrm{Spec}}(A)$ and let $\mathfrak m \subset A$ be the maximal ideal corresponding to $x$. Set $Z = X \setminus U$ and $Z' = Z \cup \{ x\} $. By Schemes, Lemma 25.12.4 there are quasi-coherent sheaves of ideals $\mathcal{I}$, resp. $\mathcal{I}'$ cutting out the reduced closed subschemes $Z$, resp. $Z'$. Consider the short exact sequence

\[ 0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0. \]

Since $x$ is a closed point of $X$ and $x \not\in Z$ we see that $\mathcal{I}/\mathcal{I}'$ is supported at $x$. In fact, the restriction of $\mathcal{I}/\mathcal{I'}$ to $U$ corresponds to the $A$-module $A/\mathfrak m$. Hence we see that $\Gamma (X, \mathcal{I}/\mathcal{I'}) = A/\mathfrak m$. Since by assumption $H^1(X, \mathcal{I}') = 0$ we see there exists a global section $f \in \Gamma (X, \mathcal{I})$ which maps to the element $1 \in A/\mathfrak m$ as a section of $\mathcal{I}/\mathcal{I'}$. Clearly we have $x \in X_ f \subset U$. This implies that $X_ f = D(f_ A)$ where $f_ A$ is the image of $f$ in $A = \Gamma (U, \mathcal{O}_ X)$. In particular $X_ f$ is affine.

Consider the union $W = \bigcup X_ f$ over all $f \in \Gamma (X, \mathcal{O}_ X)$ such that $X_ f$ is affine. Obviously $W$ is open in $X$. By the arguments above every closed point of $X$ is contained in $W$. The closed subset $X \setminus W$ of $X$ is also quasi-compact (see Topology, Lemma 5.12.3). Hence it has a closed point if it is nonempty (see Topology, Lemma 5.12.8). This would contradict the fact that all closed points are in $W$. Hence we conclude $X = W$.

Choose finitely many $f_1, \ldots , f_ n \in \Gamma (X, \mathcal{O}_ X)$ such that $X = X_{f_1} \cup \ldots \cup X_{f_ n}$ and such that each $X_{f_ i}$ is affine. This is possible as we've seen above. By Properties, Lemma 27.27.3 to finish the proof it suffices to show that $f_1, \ldots , f_ n$ generate the unit ideal in $\Gamma (X, \mathcal{O}_ X)$. Consider the short exact sequence

\[ \xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{O}_ X^{\oplus n} \ar[rr]^{f_1, \ldots , f_ n} & & \mathcal{O}_ X \ar[r] & 0 } \]

The arrow defined by $f_1, \ldots , f_ n$ is surjective since the opens $X_{f_ i}$ cover $X$. We let $\mathcal{F}$ be the kernel of this surjective map. Observe that $\mathcal{F}$ has a filtration

\[ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ n = \mathcal{F} \]

so that each subquotient $\mathcal{F}_ i/\mathcal{F}_{i - 1}$ is isomorphic to a quasi-coherent sheaf of ideals. Namely we can take $\mathcal{F}_ i$ to be the intersection of $\mathcal{F}$ with the first $i$ direct summands of $\mathcal{O}_ X^{\oplus n}$. The assumption of the lemma implies that $H^1(X, \mathcal{F}_ i/\mathcal{F}_{i - 1}) = 0$ for all $i$. This implies that $H^1(X, \mathcal{F}_2) = 0$ because it is sandwiched between $H^1(X, \mathcal{F}_1)$ and $H^1(X, \mathcal{F}_2/\mathcal{F}_1)$. Continuing like this we deduce that $H^1(X, \mathcal{F}) = 0$. Therefore we conclude that the map

\[ \xymatrix{ \bigoplus \nolimits _{i = 1, \ldots , n} \Gamma (X, \mathcal{O}_ X) \ar[rr]^{f_1, \ldots , f_ n} & & \Gamma (X, \mathcal{O}_ X) } \]

is surjective as desired. $\square$

Note that if $X$ is a Noetherian scheme then every quasi-coherent sheaf of ideals is automatically a coherent sheaf of ideals and a finite type quasi-coherent sheaf of ideals. Hence the preceding lemma and the next lemma both apply in this case.

sloganreference

Lemma 29.3.2. Let $X$ be a scheme. Assume that

  1. $X$ is quasi-compact,

  2. $X$ is quasi-separated, and

  3. $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals $\mathcal{I}$ of finite type.

Then $X$ is affine.

Proof. By Properties, Lemma 27.22.3 every quasi-coherent sheaf of ideals is a directed colimit of quasi-coherent sheaves of ideals of finite type. By Cohomology, Lemma 20.20.1 taking cohomology on $X$ commutes with directed colimits. Hence we see that $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals on $X$. In other words we see that Lemma 29.3.1 applies. $\square$

We can use the arguments given above to find a sufficient condition to see when an invertible sheaf is ample. However, we warn the reader that this condition is not necessary.

Lemma 29.3.3. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume that

  1. $X$ is quasi-compact,

  2. for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) = 0$.

Then $\mathcal{L}$ is ample.

Proof. This is proved in exactly the same way as Lemma 29.3.1. Let $x \in X$ be a closed point. Let $U \subset X$ be an affine open neighbourhood of $x$ such that $\mathcal{L}|_ U \cong \mathcal{O}_ U$. Write $U = \mathop{\mathrm{Spec}}(A)$ and let $\mathfrak m \subset A$ be the maximal ideal corresponding to $x$. Set $Z = X \setminus U$ and $Z' = Z \cup \{ x\} $. By Schemes, Lemma 25.12.4 there are quasi-coherent sheaves of ideals $\mathcal{I}$, resp. $\mathcal{I}'$ cutting out the reduced closed subschemes $Z$, resp. $Z'$. Consider the short exact sequence

\[ 0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0. \]

For every $n \geq 1$ we obtain a short exact sequence

\[ 0 \to \mathcal{I}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} \to \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} \to \mathcal{I}/\mathcal{I}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} \to 0. \]

By our assumption we may pick $n$ such that $H^1(X, \mathcal{I}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) = 0$. Since $x$ is a closed point of $X$ and $x \not\in Z$ we see that $\mathcal{I}/\mathcal{I}'$ is supported at $x$. In fact, the restriction of $\mathcal{I}/\mathcal{I'}$ to $U$ corresponds to the $A$-module $A/\mathfrak m$. Since $\mathcal{L}$ is trivial on $U$ we see that the restriction of $\mathcal{I}/\mathcal{I}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ to $U$ also corresponds to the $A$-module $A/\mathfrak m$. Hence we see that $\Gamma (X, \mathcal{I}/\mathcal{I'} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) = A/\mathfrak m$. By our choice of $n$ we see there exists a global section $s \in \Gamma (X, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$ which maps to the element $1 \in A/\mathfrak m$. Clearly we have $x \in X_ s \subset U$ because $s$ vanishes at points of $Z$. This implies that $X_ s = D(f)$ where $f \in A$ is the image of $s$ in $A \cong \Gamma (U, \mathcal{L}^{\otimes n})$. In particular $X_ s$ is affine.

Consider the union $W = \bigcup X_ s$ over all $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ for $n \geq 1$ such that $X_ s$ is affine. Obviously $W$ is open in $X$. By the arguments above every closed point of $X$ is contained in $W$. The closed subset $X \setminus W$ of $X$ is also quasi-compact (see Topology, Lemma 5.12.3). Hence it has a closed point if it is nonempty (see Topology, Lemma 5.12.8). This would contradict the fact that all closed points are in $W$. Hence we conclude $X = W$. This means that $\mathcal{L}$ is ample by Properties, Definition 27.26.1. $\square$

There is a variant of Lemma 29.3.3 with finite type ideal sheaves which we will formulate and prove here if we ever need it.


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Comment #932 by correction_bot on

"This is a would contradict the fact that all…", delete "is a".


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