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A regular global section of an invertible sheaf glues to an effective Cartier divisor

Statement

Let $X$ be a scheme, let $L$ be an invertible $O_{X}$ -module (Invertible sheaves) and let $s \in Γ (X, L)$ be a global section. Let ${U_{i}}_{i \in I}$ be an open cover of $X$ with generators $e_{i} \in L (U_{i})$ , so that $O_{U_{i}} \to L ∣_{U_{i}}$ , $a \mapsto a e_{i}$ , is an isomorphism, and let $f_{i} \in O_{X} (U_{i})$ be the coefficient defined by $s ∣_{U_{i}} = f_{i} e_{i}$ . Suppose that $s$ is a regular section of $L$ , meaning that the morphism of sheaves $O_{X} \to L$ , $a \mapsto a s$ , is injective; equivalently, suppose that each coefficient $f_{i}$ is a regular section of $O_{U_{i}}$ (Sheaf total quotient rings), that is, multiplication by the germ $(f_{i})_{x}$ is injective on $O_{X, x}$ for every $x \in U_{i}$ . Then:

(independence) the coefficient $f_{i}$ of $s$ is regular for every trivializing open cover and every choice of generators, so the hypothesis is a property of $s$ alone;
(gluing) $f_{i} / f_{j} \in O_{X} (U_{i} \cap U_{j})^{\times}$ for all $i, j$ , so the equations $f_{i}$ glue to an effective Cartier divisor $D$ on $X$ with local-equation datum ${(U_{i}, f_{i})}$ and vanishing subscheme $Z_{D}$ (Effective cartier divisor, Effective Cartier divisors are closed subschemes cut out by regular equations);
(the pair) the local isomorphisms $O_{X} (D) ∣_{U_{i}} \to L ∣_{U_{i}}$ with $f_{i}^{- 1} \mapsto e_{i}$ glue to a canonical isomorphism $φ : O_{X} (D) \to L$ (Invertible sheaf of cartier divisor, The sheaf of a Cartier divisor is invertible) which carries the canonical global section $1_{D}$ of $O_{X} (D)$ to $s$ ; consequently $(L, s) ≅ (O_{X} (D), 1_{D})$ ;
(zero locus) the ideal sheaf $I_{D}$ of $Z_{D}$ satisfies $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ ; that is, $Z_{D}$ is locally cut out by the coefficient of $s$ in each trivialization.

No integrality, reducedness or Noetherian hypothesis is imposed on $X$ .

Facts & Assumptions

Given: A scheme $X$ , an invertible $O_{X}$ -module $L$ (Invertible sheaves), a global section $s \in Γ (X, L)$ , an open cover ${U_{i}}_{i \in I}$ of $X$ with generators $e_{i} \in L (U_{i})$ , and coefficients $f_{i} \in O_{X} (U_{i})$ with $s ∣_{U_{i}} = f_{i} e_{i}$ .

[F1]

$L$ is locally free of rank one: each generator $e_{i}$ induces an isomorphism $O_{U_{i}} \to L ∣_{U_{i}}$ , $a \mapsto a e_{i}$ ; if two sections $e, e^{'}$ generate $L$ on an open $V$ then $e^{'} = w e$ for a unique $w \in O_{X} (V)$ , and $w$ is a unit of $O_{X} (V)$ , while any unit multiple of a generator is again a generator (Invertible sheaves).

[F2]

With $S_{X} (U)$ the set of regular sections of $O_{X}$ over $U$ , the sheaf $K_{X}$ of meromorphic functions is the sheafification of $U \mapsto S_{X} (U)^{- 1} O_{X} (U)$ ; the canonical maps $O_{X} (U) \to S_{X} (U)^{- 1} O_{X} (U) \to K_{X} (U)$ are ring maps that send every element of $S_{X} (U)$ to a unit, and $O_{X} \to K_{X}$ is injective. A germ is regular exactly when multiplication by it is injective (Sheaf total quotient rings).

[F3]

A Cartier divisor on $X$ is a global section of $K_{X}^{\times} / O_{X}^{\times}$ , represented by meromorphic units $f_{i} \in K_{X} (U_{i})^{\times}$ on an open cover with $f_{i} / f_{j} \in O_{X} (U_{i} \cap U_{j})^{\times}$ ; it is effective when it admits such a representation with $f_{i} \in O_{X} (U_{i})$ regular (Cartier divisor, Effective cartier divisor).

[F4]

For a Cartier divisor $D$ with datum $(U_{i}, f_{i})$ the sheaf $O_{X} (D) \subseteq K_{X}$ satisfies $O_{X} (D) ∣_{U_{i}} = f_{i}^{- 1} O_{U_{i}}$ and is invertible, freely generated by $f_{i}^{- 1}$ ; if $D$ is effective, the constant meromorphic function $1$ is a global section of $O_{X} (D)$ , called the canonical section $1_{D}$ , and $1_{D} ∣_{U_{i}} = f_{i} \cdot f_{i}^{- 1}$ (Invertible sheaf of cartier divisor, The sheaf of a Cartier divisor is invertible).

[F5]

An effective Cartier divisor $D$ determines a closed immersion $i_{D} : Z_{D} ↪ X$ whose ideal sheaf $I_{D}$ satisfies $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ for every local-equation datum $(U_{i}, f_{i})$ of $D$ (Effective Cartier divisors are closed subschemes cut out by regular equations).

[F6]

Sections of a sheaf, and morphisms of sheaves, that agree on the members of an open cover glue uniquely (A sheaf on a topological space).

Proof

1.1F1F2

Coefficients and regularity. Fix $i$ and a point $x \in U_{i}$ . Since $e_{i}$ generates $L$ on $U_{i}$ , the map $O_{X, x} \to L_{x}$ , $a \mapsto a e_{i, x}$ , is an isomorphism by [F1], and the germ of the structure map $O_{X} \to L$ , $a \mapsto a s$ , at $x$ is the composite of $a \mapsto a f_{i, x}$ with it. Hence the structure map is injective at $x$ if and only if multiplication by $f_{i, x}$ on $O_{X, x}$ is injective, i.e. if and only if the coefficient $f_{i}$ is regular at $x$ in the sense of [F2]. Therefore $s$ is a regular section of $L$ exactly when every coefficient $f_{i}$ is a regular section of $O_{U_{i}}$ .

2.1F1F3step 1.1

Unit ratios. For each pair $i, j$ both $e_{i}$ and $e_{j}$ generate $L$ over $U_{i} \cap U_{j}$ , so $e_{i} = u_{i j} e_{j}$ for a unique unit $u_{i j} \in O_{X} (U_{i} \cap U_{j})^{\times}$ by [F1]. Restricting $s ∣_{U_{i}} = f_{i} e_{i}$ and $s ∣_{U_{j}} = f_{j} e_{j}$ to the overlap and substituting gives $f_{i} u_{i j} e_{j} = f_{j} e_{j}$ , and since $a \mapsto a e_{j}$ is injective on $U_{i} \cap U_{j}$ by [F1] we get $f_{i} u_{i j} = f_{j}$ , that is, $f_{i} / f_{j} = u_{i j}^{- 1} \in O_{X} (U_{i} \cap U_{j})^{\times}$ . The equations therefore have unit ratios and, by [F3], their images in $K_{X}$ are meromorphic units with unit ratios on overlaps.

3.1F1step 1.1step 2.1

Independence of the trivialization. Let ${V_{k}}_{k \in J}$ be any other open cover with generators $e_{k}^{'} \in L (V_{k})$ and coefficients $h_{k} \in O_{X} (V_{k})$ , $s ∣_{V_{k}} = h_{k} e_{k}^{'}$ , and suppose the $f_{i}$ are regular. Fix $k$ and a point $x \in V_{k}$ ; choose $i$ with $x \in U_{i}$ . On $U_{i} \cap V_{k}$ both $e_{i}$ and $e_{k}^{'}$ generate $L$ , so $e_{k}^{'} = w e_{i}$ with $w$ a unit of $O_{X} (U_{i} \cap V_{k})$ by [F1], and comparing coefficients as in step 2.1 gives $h_{k} w = f_{i}$ , so $h_{k} ∣_{U_{i} \cap V_{k}} = w^{- 1} f_{i} ∣_{U_{i} \cap V_{k}}$ is a unit multiple of the regular section $f_{i} ∣_{U_{i} \cap V_{k}}$ , hence is regular. As $x \in V_{k}$ was arbitrary, $h_{k}$ is a regular section of $O_{V_{k}}$ ; thus regularity of the coefficients is independent of the cover and of the chosen generators, and by step 1.1 it is equivalent to regularity of $s$ .

3.2F2F3step 2.1

The effective Cartier divisor. By the hypothesis of the Statement (equivalently, by steps 1.1 and 3.1) the coefficients $f_{i}$ are regular sections of $O_{U_{i}}$ , hence their images in $K_{X} (U_{i})$ are units by [F2], and their ratios are units of $O_{X}$ on the overlaps by step 2.1. Therefore $(U_{i}, f_{i})$ is a local-equation datum of a Cartier divisor $D$ on $X$ in the sense of [F3], and $D$ is effective because the representing equations lie in $O_{X} (U_{i})$ and are regular.

3.3F1F4F6step 2.1

The glued isomorphism. By [F4] the sheaf $O_{X} (D)$ is freely generated by $f_{i}^{- 1}$ on each $U_{i}$ , and $L$ is freely generated by $e_{i}$ there; let $φ_{i} : O_{X} (D) ∣_{U_{i}} \to L ∣_{U_{i}}$ be the unique $O_{U_{i}}$ -linear isomorphism with $φ_{i} (f_{i}^{- 1}) = e_{i}$ . On $U_{i} \cap U_{j}$ we have $f_{i}^{- 1} = u_{i j} f_{j}^{- 1}$ and $e_{i} = u_{i j} e_{j}$ by step 2.1, so $φ_{i} (f_{j}^{- 1}) = φ_{i} (u_{i j}^{- 1} f_{i}^{- 1}) = u_{i j}^{- 1} e_{i} = e_{j} = φ_{j} (f_{j}^{- 1})$ ; the two isomorphisms agree on a generator, hence on all sections. By the gluing axiom [F6] the $φ_{i}$ glue to a morphism $φ : O_{X} (D) \to L$ , which is an isomorphism because it restricts to an isomorphism on each member of the cover.

4.1F4F5F6step 3.3

The canonical section realizes $s$ . The constant meromorphic function $1$ restricts to $f_{i} \cdot f_{i}^{- 1} \in O_{X} (D) (U_{i})$ for every $i$ by [F4], so it is the global section $1_{D} \in Γ (X, O_{X} (D))$ and these local expressions glue. Under $φ$ its restriction to $U_{i}$ maps to $f_{i} e_{i} = s ∣_{U_{i}}$ by step 3.3; the local sections $s ∣_{U_{i}}$ agree on overlaps because they are restrictions of $s$ , so $φ (1_{D}) = s$ by the uniqueness part of [F6]. Hence $(L, s) ≅ (O_{X} (D), 1_{D})$ , and by [F5] the ideal sheaf of the vanishing subscheme satisfies $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ , i.e. $Z_{D}$ is locally cut out by the coefficient of $s$ in each trivialization.

5.1step 3.1step 3.2step 3.3step 4.1∎

Conclusion. Every regular global section $s$ of an invertible sheaf $L$ has regular coefficients on every trivialization (steps 1.1 and 3.1); those coefficients have unit ratios and glue to an effective Cartier divisor $D$ (steps 2.1 and 3.2); the local isomorphisms on the free generators glue to a canonical isomorphism $φ : O_{X} (D) \to L$ with $φ (1_{D}) = s$ (steps 3.3 and 4.1); and $Z_{D}$ is locally cut out by the coefficients of $s$ (step 4.1). This proves all four assertions of the Statement.

The construction uses no choice principle: the cover, the generators and the section are given, the coefficients and the glueing isomorphisms are uniquely determined by them, and no trivialization or divisor is selected. The zero divisor arises exactly from unit coefficients: if $f_{i} = 1$ for all $i$ then $s$ generates $L$ , the divisor $D$ is the empty effective divisor with $Z_{D} = \emptyset$ and $O_{X} (D) = O_{X}$ , and $φ$ is the inverse of the given trivialization. Conversely a regular section is never identically zero on a nonempty open: if $U \neq \emptyset$ then injectivity of $O_{X} (U) \to L (U)$ forces $s ∣_{U} \neq 0$ , so the zero section is regular only on the empty scheme. On the empty scheme the cover is empty, the data are vacuous, $D = 0$ , $O_{X} (D) = O_{X}$ is the zero module sheaf, which is invertible there, and the canonical isomorphism is the identity. The statement is local in $X$ : it applies to a section defined on any open subscheme, and it imposes no condition on the singularities, the reducedness or the integrality of $X$ ; a coefficient may be any regular section of $O_{X}$ , including a non-zero-divisor that vanishes at a closed point of a nonreduced scheme.

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Sources

The Stacks Project, Divisors, Definition 31.15.6 and Lemma 31.15.10 (tags 01WY and 01X0): regular sections of invertible sheaves and the bijection with effective Cartier divisors (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Ch. 15 Exercises 15.3.B-15.3.C and Section 15.6 (canonical sections and effective Cartier divisors) (standard reference, not scraped)