## 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. $\square$

Comment #6696 by old friend on

In Definition 01WR, it might be better to use exactly the same language in both (1) and (2) for expository purposes e.g. in (1), say "... whose ideal sheaf I_{D} subset O_{S} is locally generated by a single element". It helps in seeing the difference between the two parts.

Comment #6902 by on

Well, we're trying to actually distinguish between the two cases by calling effective Cartier divisors always $D$ and locally principal closed subschemes are $Z$ or $Y$ or something. Anyway, I think it is OK like this for now. If others second your suggestion, I will change it.

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