The Stacks project

31.13 Effective Cartier divisors

We define the notion of an effective Cartier divisor before any other type of divisor.

Definition 31.13.1. Let $S$ be a scheme.

  1. A locally principal closed subscheme of $S$ is a closed subscheme whose sheaf of ideals is locally generated by a single element.

  2. An effective Cartier divisor on $S$ is a closed subscheme $D \subset S$ whose ideal sheaf $\mathcal{I}_ D \subset \mathcal{O}_ S$ is an invertible $\mathcal{O}_ S$-module.

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

Lemma 31.13.2. Let $S$ be a scheme. Let $D \subset S$ be a closed subscheme. The following are equivalent:

  1. The subscheme $D$ is an effective Cartier divisor on $S$.

  2. For every $x \in D$ there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset S$ of $x$ such that $U \cap D = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ a nonzerodivisor.

Proof. Assume (1). For every $x \in D$ there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset S$ of $x$ such that $\mathcal{I}_ D|_ U \cong \mathcal{O}_ U$. In other words, there exists a section $f \in \Gamma (U, \mathcal{I}_ D)$ which freely generates the restriction $\mathcal{I}_ D|_ U$. Hence $f \in A$, and the multiplication map $f : A \to A$ is injective. Also, since $\mathcal{I}_ D$ is quasi-coherent we see that $D \cap U = \mathop{\mathrm{Spec}}(A/(f))$.

Assume (2). Let $x \in D$. By assumption there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset S$ of $x$ such that $U \cap D = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ a nonzerodivisor. Then $\mathcal{I}_ D|_ U \cong \mathcal{O}_ U$ since it is equal to $\widetilde{(f)} \cong \widetilde{A} \cong \mathcal{O}_ U$. Of course $\mathcal{I}_ D$ restricted to the open subscheme $S \setminus D$ is isomorphic to $\mathcal{O}_{S \setminus D}$. Hence $\mathcal{I}_ D$ is an invertible $\mathcal{O}_ S$-module. $\square$

Lemma 31.13.3. Let $S$ be a scheme. Let $Z \subset S$ be a locally principal closed subscheme. Let $U = S \setminus Z$. Then $U \to S$ is an affine morphism.

Proof. The question is local on $S$, see Morphisms, Lemmas 29.11.3. Thus we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $Z = V(f)$ for some $f \in A$. In this case $U = D(f) = \mathop{\mathrm{Spec}}(A_ f)$ is affine hence $U \to S$ is affine. $\square$

Lemma 31.13.4. Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor. Let $U = S \setminus D$. Then $U \to S$ is an affine morphism and $U$ is scheme theoretically dense in $S$.

Proof. Affineness is Lemma 31.13.3. The density question is local on $S$, see Morphisms, Lemma 29.7.5. Thus we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $D$ corresponding to the nonzerodivisor $f \in A$, see Lemma 31.13.2. Thus $A \subset A_ f$ which implies that $U \subset S$ is scheme theoretically dense, see Morphisms, Example 29.7.4. $\square$

Lemma 31.13.5. Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor. Let $s \in D$. If $\dim _ s(S) < \infty $, then $\dim _ s(D) < \dim _ s(S)$.

Proof. Assume $\dim _ s(S) < \infty $. Let $U = \mathop{\mathrm{Spec}}(A) \subset S$ be an affine open neighbourhood of $s$ such that $\dim (U) = \dim _ s(S)$ and such that $D = V(f)$ for some nonzerodivisor $f \in A$ (see Lemma 31.13.2). Recall that $\dim (U)$ is the Krull dimension of the ring $A$ and that $\dim (U \cap D)$ is the Krull dimension of the ring $A/(f)$. Then $f$ is not contained in any minimal prime of $A$. Hence any maximal chain of primes in $A/(f)$, viewed as a chain of primes in $A$, can be extended by adding a minimal prime. $\square$

Definition 31.13.6. Let $S$ be a scheme. Given effective Cartier divisors $D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the closed subscheme of $S$ 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 31.13.7. The sum of two effective Cartier divisors is an effective Cartier divisor.

Proof. Omitted. Locally $f_1, f_2 \in A$ are nonzerodivisors, then also $f_1f_2 \in A$ is a nonzerodivisor. $\square$

Lemma 31.13.8. Let $X$ be a scheme. Let $D, D'$ be two effective Cartier divisors on $X$. If $D \subset D'$ (as closed subschemes of $X$), then there exists an effective Cartier divisor $D''$ such that $D' = D + D''$.

Proof. Omitted. $\square$

Lemma 31.13.9. Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes 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. Applying Lemma 31.13.2 we obtain the following algebra situation: $A$ is a ring, $I, J \subset A$ ideals and $f \in A$ a nonzerodivisor such that $IJ = (f)$. Thus the result follows from Algebra, Lemma 10.120.16. $\square$

Lemma 31.13.10. Let $X$ be a scheme. Let $D, D' \subset X$ be effective Cartier divisors such that the scheme theoretic intersection $D \cap D'$ is an effective Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic union of $D$ and $D'$.

Proof. See Morphisms, Definition 29.4.4 for the definition of scheme theoretic intersection and union. To prove the lemma working locally (using Lemma 31.13.2) we obtain the following algebra problem: Given a ring $A$ and nonzerodivisors $f_1, f_2 \in A$ such that $f_1$ maps to a nonzerodivisor in $A/f_2A$, show that $f_1A \cap f_2A = f_1f_2A$. We omit the straightforward argument. $\square$

Recall that we have defined the inverse image of a closed subscheme under any morphism of schemes in Schemes, Definition 26.17.7.

Lemma 31.13.11. Let $f : S' \to S$ be a morphism of schemes. Let $Z \subset S$ be a locally principal closed subscheme. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subscheme of $S'$.

Proof. Omitted. $\square$

Definition 31.13.12. Let $f : S' \to S$ be a morphism of schemes. Let $D \subset S$ be an effective Cartier divisor. We say the pullback of $D$ by $f$ is defined if the closed subscheme $f^{-1}(D) \subset S'$ 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. Here is an example lemma.

Lemma 31.13.13. Let $f : X \to Y$ be a morphism of schemes. 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. $X$, $Y$ integral and $f$ dominant,

  3. $X$ reduced and $f(\xi ) \not\in D$ for any generic point $\xi $ of any irreducible component of $X$,

  4. $X$ is locally Noetherian and $f(x) \not\in D$ for any associated point $x$ of $X$,

  5. $X$ is locally Noetherian, has no embedded points, and $f(\xi ) \not\in D$ for any generic point $\xi $ of an irreducible component of $X$,

  6. $f$ is flat, and

  7. add more here as needed.

Proof. The question is local on $X$, and hence we reduce to the case where $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(R)$, $f$ is given by $\varphi : R \to A$ and $D = \mathop{\mathrm{Spec}}(R/(t))$ where $t \in R$ is a nonzerodivisor. The goal in each case is to show that $\varphi (t) \in A$ is a nonzerodivisor.

In case (1) this follows from Algebra, Lemma 10.66.7. Case (4) is a special case of (1) by Lemma 31.5.8. Case (5) follows from (4) and the definitions. Case (3) is a special case of (1) by Lemma 31.5.12. Case (2) is a special case of (3). If $R \to A$ is flat, then $t : R \to R$ being injective shows that $t : A \to A$ is injective. This proves (6). $\square$

Lemma 31.13.14. Let $f : S' \to S$ be a morphism of schemes. Let $D_1$, $D_2$ be effective Cartier divisors on $S$. 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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