The Stacks project

68.6 Effective Cartier divisors

For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Note that in Morphisms of Spaces, Section 64.13 we discussed the correspondence between closed subspaces and quasi-coherent sheaves of ideals. Moreover, in Properties of Spaces, Section 63.30, we discussed properties of quasi-coherent modules, in particular “locally generated by $1$ element”. These references show that the following definition is compatible with the definition for schemes.

Definition 68.6.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. A locally principal closed subspace of $X$ is a closed subspace whose sheaf of ideals is locally generated by $1$ element.

  2. An effective Cartier divisor on $X$ is a closed subspace $D \subset X$ such that the ideal sheaf $\mathcal{I}_ D \subset \mathcal{O}_ X$ is an invertible $\mathcal{O}_ X$-module.

Thus an effective Cartier divisor is a locally principal closed subspace, but the converse is not always true. Effective Cartier divisors are closed subspaces of pure codimension $1$ in the strongest possible sense. Namely they are locally cut out by a single element which is not a zerodivisor. In particular they are nowhere dense.

Lemma 68.6.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be a closed subspace. The following are equivalent:

  1. The subspace $D$ is an effective Cartier divisor on $X$.

  2. For some scheme $U$ and surjective étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.

  3. For every scheme $U$ and every étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.

  4. For every $x \in |D|$ there exists an étale morphism $(U, u) \to (X, x)$ of pointed algebraic spaces such that $U = \mathop{\mathrm{Spec}}(A)$ and $D \times _ X U = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ not a zerodivisor.

Proof. The equivalence of (1) – (3) follows from Definition 68.6.1 and the references preceding it. Assume (1) and let $x \in |D|$. Choose a scheme $W$ and a surjective étale morphism $W \to X$. Choose $w \in D \times _ X W$ mapping to $x$. By (3) $D \times _ X W$ is an effective Cartier divisor on $W$. Hence we can find affine étale neighbourhood $U$ by choosing an affine open neighbourhood of $w$ in $W$ as in Divisors, Lemma 31.13.2.

Assume (4). Then we see that $\mathcal{I}_ D|_ U$ is invertible by Divisors, Lemma 31.13.2. Since we can find an étale covering of $X$ by the collection of all such $U$ and $X \setminus D$, we conclude that $\mathcal{I}_ D$ is an invertible $\mathcal{O}_ X$-module. $\square$

Lemma 68.6.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a locally principal closed subspace. Let $U = X \setminus Z$. Then $U \to X$ is an affine morphism.

Proof. The question is étale local on $X$, see Morphisms of Spaces, Lemmas 64.20.3 and Lemma 68.6.2. Thus this follows from the case of schemes which is Divisors, Lemma 31.13.3. $\square$

Lemma 68.6.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $U = X \setminus D$. Then $U \to X$ is an affine morphism and $U$ is scheme theoretically dense in $X$.

Proof. Affineness is Lemma 68.6.3. The density question is étale local on $X$ by Morphisms of Spaces, Definition 64.17.3. Thus this follows from the case of schemes which is Divisors, Lemma 31.13.4. $\square$

Lemma 68.6.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $x \in |D|$. If $\dim _ x(X) < \infty $, then $\dim _ x(D) < \dim _ x(X)$.

Proof. Both the definition of an effective Cartier divisor and of the dimension of an algebraic space at a point (Properties of Spaces, Definition 63.9.1) are étale local. Hence this lemma follows from the case of schemes which is Divisors, Lemma 31.13.5. $\square$

Definition 68.6.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Given effective Cartier divisors $D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the closed subspace of $X$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I}_{D_1}\mathcal{I}_{D_2} \subset \mathcal{O}_ S$. We call this the sum of the effective Cartier divisors $D_1$ and $D_2$.

It is clear that we may define the sum $\sum n_ iD_ i$ given finitely many effective Cartier divisors $D_ i$ on $X$ and nonnegative integers $n_ i$.

Lemma 68.6.7. The sum of two effective Cartier divisors is an effective Cartier divisor.

Proof. Omitted. Étale locally this reduces to the following simple algebra fact: if $f_1, f_2 \in A$ are nonzerodivisors of a ring $A$, then $f_1f_2 \in A$ is a nonzerodivisor. $\square$

Lemma 68.6.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y$ be two closed subspaces of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$.

Proof. By Lemma 68.6.2 this reduces to the case of schemes which is Divisors, Lemma 31.13.9. $\square$

Recall that we have defined the inverse image of a closed subspace under any morphism of algebraic spaces in Morphisms of Spaces, Definition 64.13.2.

Lemma 68.6.9. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a locally principal closed subspace. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subspace of $X'$.

Proof. Omitted. $\square$

Definition 68.6.10. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D \subset X$ be an effective Cartier divisor. We say the pullback of $D$ by $f$ is defined if the closed subspace $f^{-1}(D) \subset X'$ is an effective Cartier divisor. In this case we denote it either $f^*D$ or $f^{-1}(D)$ and we call it the pullback of the effective Cartier divisor.

The condition that $f^{-1}(D)$ is an effective Cartier divisor is often satisfied in practice.

Lemma 68.6.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $D \subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases:

  1. $f(x) \not\in |D|$ for any weakly associated point $x$ of $X$,

  2. $f$ is flat, and

  3. add more here as needed.

Proof. Working étale locally this lemma reduces to the case of schemes, see Divisors, Lemma 31.13.13. $\square$

Lemma 68.6.12. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$.

Proof. Omitted. $\square$


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