The Stacks project

71.7 Effective Cartier divisors and invertible sheaves

Since an effective Cartier divisor has an invertible ideal sheaf (Definition 71.6.1) the following definition makes sense.

Definition 71.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I}_ D$.

  1. The invertible sheaf $\mathcal{O}_ X(D)$ associated to $D$ is defined by

    \[ \mathcal{O}_ X(D) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}_ D, \mathcal{O}_ X) = \mathcal{I}_ D^{\otimes -1}. \]
  2. The canonical section, usually denoted $1$ or $1_ D$, is the global section of $\mathcal{O}_ X(D)$ corresponding to the inclusion mapping $\mathcal{I}_ D \to \mathcal{O}_ X$.

  3. We write $\mathcal{O}_ X(-D) = \mathcal{O}_ X(D)^{\otimes -1} = \mathcal{I}_ D$.

  4. Given a second effective Cartier divisor $D' \subset X$ we define $\mathcal{O}_ X(D - D') = \mathcal{O}_ X(D) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(-D')$.

Some comments. We will see below that the assignment $D \mapsto \mathcal{O}_ X(D)$ turns addition of effective Cartier divisors (Definition 71.6.6) into addition in the Picard group of $X$ (Lemma 71.7.3). However, the expression $D - D'$ in the definition above does not have any geometric meaning. More precisely, we can think of the set of effective Cartier divisors on $X$ as a commutative monoid $\text{EffCart}(X)$ whose zero element is the empty effective Cartier divisor. Then the assignment $(D, D') \mapsto \mathcal{O}_ X(D - D')$ defines a group homomorphism

\[ \text{EffCart}(X)^{gp} \longrightarrow \mathop{\mathrm{Pic}}\nolimits (X) \]

where the left hand side is the group completion of $\text{EffCart}(X)$. In other words, when we write $\mathcal{O}_ X(D - D')$ we may think of $D - D'$ as an element of $\text{EffCart}(X)^{gp}$.

Lemma 71.7.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Then for the conormal sheaf we have $\mathcal{C}_{D/X} = \mathcal{I}_ D|D = \mathcal{O}_ X(D)^{\otimes -1}|_ D$.

Proof. Omitted. $\square$

Lemma 71.7.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. Let $D = D_1 + D_2$. Then there is a unique isomorphism

\[ \mathcal{O}_ X(D_1) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(D_2) \longrightarrow \mathcal{O}_ X(D) \]

which maps $1_{D_1} \otimes 1_{D_2}$ to $1_ D$.

Proof. Omitted. $\square$

Definition 71.7.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf on $X$. A global section $s \in \Gamma (X, \mathcal{L})$ is called a regular section if the map $\mathcal{O}_ X \to \mathcal{L}$, $f \mapsto fs$ is injective.

Lemma 71.7.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f \in \Gamma (X, \mathcal{O}_ X)$. The following are equivalent:

  1. $f$ is a regular section, and

  2. for any $x \in X$ the image $f \in \mathcal{O}_{X, \overline{x}}$ is not a zerodivisor.

  3. for any affine $U = \mathop{\mathrm{Spec}}(A)$ ├ętale over $X$ the restriction $f|_ U$ is a nonzerodivisor of $A$, and

  4. there exists a scheme $U$ and a surjective ├ętale morphism $U \to X$ such that $f|_ U$ is a regular section of $\mathcal{O}_ U$.

Proof. Omitted. $\square$

Note that a global section $s$ of an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ may be seen as an $\mathcal{O}_ X$-module map $s : \mathcal{O}_ X \to \mathcal{L}$. Its dual is therefore a map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_ X$. (See Modules on Sites, Lemma 18.32.4 for the dual invertible sheaf.)

Definition 71.7.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf. Let $s \in \Gamma (X, \mathcal{L})$. The zero scheme of $s$ is the closed subspace $Z(s) \subset X$ defined by the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ which is the image of the map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_ X$.

Lemma 71.7.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$.

  1. Consider closed immersions $i : Z \to X$ such that $i^*s \in \Gamma (Z, i^*\mathcal{L}))$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set.

  2. For any morphism of algebraic spaces $f : Y \to X$ over $S$ we have $f^*s = 0$ in $\Gamma (Y, f^*\mathcal{L})$ if and only if $f$ factors through $Z(s)$.

  3. The zero scheme $Z(s)$ is a locally principal closed subspace of $X$.

  4. The zero scheme $Z(s)$ is an effective Cartier divisor on $X$ if and only if $s$ is a regular section of $\mathcal{L}$.

Proof. Omitted. $\square$

Lemma 71.7.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_ D$ of $\mathcal{O}_ X(D)$ is regular.

  2. Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_ X(D) \to \mathcal{L}$ which maps $1_ D$ to $s$.

The constructions $D \mapsto (\mathcal{O}_ X(D), 1_ D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps

\[ \left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible} \\ \mathcal{O}_ X\text{-module and a regular global section} \end{matrix} \right\} \]

Proof. Omitted. $\square$


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