31.14 Effective Cartier divisors and invertible sheaves
Since an effective Cartier divisor has an invertible ideal sheaf (Definition 31.13.1) the following definition makes sense.
Definition 31.14.1. Let S be a scheme. Let D \subset S be an effective Cartier divisor with ideal sheaf \mathcal{I}_ D.
The invertible sheaf \mathcal{O}_ S(D) associated to D is defined by
\mathcal{O}_ S(D) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{I}_ D, \mathcal{O}_ S) = \mathcal{I}_ D^{\otimes -1}.
The canonical section, usually denoted 1 or 1_ D, is the global section of \mathcal{O}_ S(D) corresponding to the inclusion mapping \mathcal{I}_ D \to \mathcal{O}_ S.
We write \mathcal{O}_ S(-D) = \mathcal{O}_ S(D)^{\otimes -1} = \mathcal{I}_ D.
Given a second effective Cartier divisor D' \subset S we define \mathcal{O}_ S(D - D') = \mathcal{O}_ S(D) \otimes _{\mathcal{O}_ S} \mathcal{O}_ S(-D').
Some comments. We will see below that the assignment D \mapsto \mathcal{O}_ S(D) turns addition of effective Cartier divisors (Definition 31.13.6) into addition in the Picard group of S (Lemma 31.14.4). 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 S as a commutative monoid \text{EffCart}(S) whose zero element is the empty effective Cartier divisor. Then the assignment (D, D') \mapsto \mathcal{O}_ S(D - D') defines a group homomorphism
\text{EffCart}(S)^{gp} \longrightarrow \mathop{\mathrm{Pic}}\nolimits (S)
where the left hand side is the group completion of \text{EffCart}(S). In other words, when we write \mathcal{O}_ S(D - D') we may think of D - D' as an element of \text{EffCart}(S)^{gp}.
Lemma 31.14.2. Let S be a scheme and let D \subset S be an effective Cartier divisor. Then the conormal sheaf is \mathcal{C}_{D/S} = \mathcal{I}_ D|_ D = \mathcal{O}_ S(-D)|_ D and the normal sheaf is \mathcal{N}_{D/S} = \mathcal{O}_ S(D)|_ D.
Proof.
This follows from Morphisms, Lemma 29.31.2.
\square
Lemma 31.14.3. Let X be a scheme. Let D, C \subset X be effective Cartier divisors with C \subset D and let D' = D + C. Then there is a short exact sequence
0 \to \mathcal{O}_ X(-D)|_ C \to \mathcal{O}_{D'} \to \mathcal{O}_ D \to 0
of \mathcal{O}_ X-modules.
Proof.
In the statement of the lemma and in the proof we use the equivalence of Morphisms, Lemma 29.4.1 to think of quasi-coherent modules on closed subschemes of X as quasi-coherent modules on X. Let \mathcal{I} be the ideal sheaf of D in D'. Then there is a short exact sequence
0 \to \mathcal{I} \to \mathcal{O}_{D'} \to \mathcal{O}_ D \to 0
because D \to D' is a closed immersion. There is a canonical surjection \mathcal{I} \to \mathcal{I}/\mathcal{I}^2 = \mathcal{C}_{D/D'}. We have \mathcal{C}_{D/X} = \mathcal{O}_ X(-D)|_ D by Lemma 31.14.2 and there is a canonical surjective map
\mathcal{C}_{D/X} \longrightarrow \mathcal{C}_{D/D'}
see Morphisms, Lemmas 29.31.3 and 29.31.4. Thus it suffices to show: (a) \mathcal{I}^2 = 0 and (b) \mathcal{I} is an invertible \mathcal{O}_ C-module. Both (a) and (b) can be checked locally, hence we may assume X = \mathop{\mathrm{Spec}}(A), D = \mathop{\mathrm{Spec}}(A/fA) and C = \mathop{\mathrm{Spec}}(A/gA) where f, g \in A are nonzerodivisors (Lemma 31.13.2). Since C \subset D we see that f \in gA. Then I = fA/fgA has square zero and is invertible as an A/gA-module as desired.
\square
Lemma 31.14.4. Let S be a scheme. Let D_1, D_2 be effective Cartier divisors on S. Let D = D_1 + D_2. Then there is a unique isomorphism
\mathcal{O}_ S(D_1) \otimes _{\mathcal{O}_ S} \mathcal{O}_ S(D_2) \longrightarrow \mathcal{O}_ S(D)
which maps 1_{D_1} \otimes 1_{D_2} to 1_ D.
Proof.
Omitted.
\square
Lemma 31.14.5. Let f : S' \to S be a morphism of schemes. Let D be a effective Cartier divisors on S. If the pullback of D is defined then f^*\mathcal{O}_ S(D) = \mathcal{O}_{S'}(f^*D) and the canonical section 1_ D pulls back to the canonical section 1_{f^*D}.
Proof.
Omitted.
\square
Definition 31.14.6. Let (X, \mathcal{O}_ X) be a locally ringed space. 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 31.14.7. Let X be a locally ringed space. Let f \in \Gamma (X, \mathcal{O}_ X). The following are equivalent:
f is a regular section, and
for any x \in X the image f \in \mathcal{O}_{X, x} is a nonzerodivisor.
If X is a scheme these are also equivalent to
for any affine open \mathop{\mathrm{Spec}}(A) = U \subset X the image f \in A is a nonzerodivisor,
there exists an affine open covering X = \bigcup \mathop{\mathrm{Spec}}(A_ i) such that the image of f in A_ i is a nonzerodivisor for all i.
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, Definition 17.25.6 for the definition of the dual invertible sheaf.)
Definition 31.14.8. Let X be a scheme. Let \mathcal{L} be an invertible sheaf. Let s \in \Gamma (X, \mathcal{L}) be a global section. The zero scheme of s is the closed subscheme 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 31.14.9. Let X be a scheme. Let \mathcal{L} be an invertible sheaf. Let s \in \Gamma (X, \mathcal{L}).
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.
For any morphism of schemes f : Y \to X we have f^*s = 0 in \Gamma (Y, f^*\mathcal{L}) if and only if f factors through Z(s).
The zero scheme Z(s) is a locally principal closed subscheme.
The zero scheme Z(s) is an effective Cartier divisor if and only if s is a regular section of \mathcal{L}.
Proof.
Omitted.
\square
Lemma 31.14.10.slogan Let X be a scheme.
If D \subset X is an effective Cartier divisor, then the canonical section 1_ D of \mathcal{O}_ X(D) is regular.
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{isomorphism classes of pairs }(\mathcal{L}, s)
\\ \text{consisting of an invertible } \mathcal{O}_ X\text{-module}
\\ \mathcal{L}\text{ and a regular global section }s
\end{matrix} \right\}
Proof.
Omitted.
\square
Comments (2)
Comment #4210 by Che Shen on
Comment #4392 by Johan on