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Effective Cartier divisors give a short exact sequence

Statement

Let $X$ be a scheme, let $D$ be an effective Cartier divisor on $X$ with closed immersion $i : D ↪ X$ and ideal sheaf $I_{D}$ as in Effective Cartier divisors are closed subschemes cut out by regular equations, and let $O_{X} (- D)$ be the associated invertible sheaf (Invertible sheaf of cartier divisor). Then there is a short exact sequence of $O_{X}$ -modules $0 ⟶ O_{X} (- D) ⟶ O_{X} ⟶ i_{*} O_{D} ⟶ 0,$ where the first map is the inclusion $I_{D} \subseteq O_{X}$ read through the identification $O_{X} (- D) = I_{D}$ , and the second is the surjection $O_{X} \to i_{*} O_{D}$ defining the closed immersion (Closed immersions of schemes, Exact sequences of sheaves).

Facts & Assumptions

Given: An effective Cartier divisor $D$ on a scheme $X$ with a local-equation datum ${(U_{i}, f_{i})}$ of regular equations (Effective cartier divisor).

[F1]

$D$ determines a closed immersion $i : D ↪ X$ whose ideal sheaf $I_{D} = \ker (O_{X} \to i_{*} O_{D})$ satisfies $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ and is invertible (Effective Cartier divisors are closed subschemes cut out by regular equations).

[F2]

For a Cartier divisor represented by the units $f_{i}$ one has $O_{X} (D) ∣_{U_{i}} = f_{i}^{- 1} O_{U_{i}}$ and $O_{X} (- D) ∣_{U_{i}} = f_{i} O_{U_{i}}$ ; for effective $D$ this last subsheaf is exactly the ideal sheaf $I_{D}$ (Invertible sheaf of cartier divisor).

[F3]

A closed immersion $i$ has surjective structure map $O_{X} \to i_{*} O_{D}$ , so $i_{*} O_{D}$ is the quotient of $O_{X}$ by the kernel of that map (Closed immersions of schemes).

[F4]

A sequence of sheaves of modules is exact when at every term the image sheaf equals the kernel sheaf (Exact sequences of sheaves).

[F5]

The kernel sheaf of a morphism is computed objectwise, and the image sheaf is the sheafification of the objectwise image (Kernel sheaves are objectwise, while cokernels and images are sheafified).

[F6]

The quotient sheaf $O_{X} / I_{D}$ is the sheafification of the presheaf $U \mapsto O_{X} (U) / I_{D} (U)$ by [F3, F5]. Its stalk at $x$ is the quotient $O_{X, x} / I_{D, x}$ . Sheafification preserves stalks (Sheafification preserves stalks), so the map from this quotient to the quotient sheaf stalk is onto: every quotient-presheaf germ is represented by a local quotient class, itself represented by a section of $O_{X}$ . Its kernel is zero: if such a section's quotient class has zero germ, it is the zero quotient class after restriction to a smaller neighbourhood by germ equality, so the section there belongs to $I_{D}$ . The given local equations satisfy $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ by [F1], so $I_{D, x} = (f_{i})_{x} O_{X, x}$ (Kernel sheaves are objectwise, while cokernels and images are sheafified, The stalk of a presheaf at a point).

[F7]

A sequence of sheaves of abelian groups is exact if and only if all its stalk sequences are exact (A sequence of abelian sheaves is exact exactly when it is exact on every stalk).

Proof

1.1F1F4F5

Chartwise exactness. On $U_{i}$ the sequence of $O_{U_{i}}$ -modules $0 \to f_{i} O_{U_{i}} ↪ O_{U_{i}} \to O_{U_{i}} / f_{i} O_{U_{i}} \to 0$ is exact. The first map is the inclusion of the ideal sheaf, hence injective, and the quotient map is surjective with kernel exactly that ideal. Multiplication by $f_{i}$ gives an isomorphism $O_{U_{i}} \to f_{i} O_{U_{i}}$ : it is injective because every germ of $f_{i}$ is a nonzerodivisor and sectionwise injectivity can be checked on stalks, and it is surjective by the definition of the principal ideal sheaf. Thus the first term is also identified with $O_{U_{i}}$ through the local equation, as required.

2.1F1F2F3step 1.1

Identifying the terms. By [F1] and [F2] we have $O_{X} (- D) ∣_{U_{i}} = f_{i} O_{U_{i}} = I_{D} ∣_{U_{i}}$ , and $O_{X} (- D) = I_{D}$ as subsheaves of $O_{X}$ because the identifications agree on overlaps. By [F3] the map $O_{X} \to i_{*} O_{D}$ is surjective with kernel $I_{D}$ , so it induces an identification $i_{*} O_{D} = O_{X} / I_{D}$ . Hence over $U_{i}$ the sequence of step 1.1 is the restriction of $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ .

3.1F6step 1.1step 2.1

Stalk sequence. Let $x \in U_{i}$ . By [F1, F2] the stalk of $I_{D}$ at $x$ is $(f_{i})_{x} O_{X, x}$ , and by [F6] the quotient map has that kernel and is surjective on the stalk. Thus the stalk sequence of the displayed sequence at $x$ is $0 \to (f_{i})_{x} O_{X, x} \to O_{X, x} \to O_{X, x} / (f_{i})_{x} O_{X, x} \to 0$ , which is exact because the first map is injective and the second has kernel exactly the image of the first.

4.1F7step 3.1∎

Conclusion. Every point of $X$ lies in some $U_{i}$ , so all stalk sequences are exact; by the stalkwise criterion the sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ is exact.

No choice principle is used: the equations are those of the given datum, and the exactness is verified stalk by stalk. For the zero effective divisor the ideal sheaf is $O_{X}$ , the closed subscheme is empty and the sequence reads $0 \to O_{X} \to O_{X} \to 0 \to 0$ ; if $X = \emptyset$ all three sheaves are the zero sheaf and the sequence is exact as well.

Depends on

Used by

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Divisors, §31.15 Remark 15.11 and Lemma 15.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Ch. 15 §§15.2–15.3 (standard reference, not scraped)