Serre's criterion for affineness.

, [II, Theorem 5.2.1 (d') and IV (1.7.17), EGA]

Lemma 30.3.1. 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 26.12.4 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 28.27.3 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. $\square$

Comment #1332 by Paul Lessard on

Affinenes in the slogan should be changed to affineness (missing the second s).

Comment #2724 by on

A reference is EGA II, Theorem 5.2.1 (d') and EGA IV_1, 1.7.17

Comment #7115 by Liu on

The assumption in EGA II (5.2.1) & EGA IV (1.7.17) says that the scheme X should be qcqs. Here we only assume quasi-compact, does it harm the argument given here?

Comment #7158 by DatPham on

@#7115: That $X$ is qcqs is in fact a consequence of the fact that $X$ has a finite cover of affine open subsets of the form $X_f$. Indeed $X_f\cap X_g=D_{X_f}(g|_{X_f})$ is affine (hence quasi-compact) as soon as $X_f,X_g$ are affine.

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