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Twisting the exact sequence of an effective Cartier divisor

Statement

Let $X$ be a scheme, let $i : D ↪ X$ be an effective Cartier divisor with ideal sheaf $I_{D}$ , so that $i$ is a closed immersion, $O_{X} (- D) = I_{D}$ is invertible and $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ is exact (Effective Cartier divisors give a short exact sequence), and let $L$ be an invertible $O_{X}$ -module (Invertible sheaves). Write $L (- D) = L \otimes_{O_{X}} O_{X} (- D), L ∣_{D} = i^{*} L$ for the twist of $L$ by $- D$ and the restriction of $L$ to $D$ (Pullback of a module along a morphism of ringed spaces). Then there is a short exact sequence of $O_{X}$ -modules $0 ⟶ L (- D) \overset{α}{\to} L \overset{β}{\to} i_{*} (L ∣_{D}) ⟶ 0,$ where $α = {id}_{L} \otimes (the inclusion I_{D} \subseteq O_{X})$ and $β$ is the tensor product of the quotient map $O_{X} \to i_{*} O_{D}$ with ${id}_{L}$ , composed with the canonical isomorphism $L \otimes_{O_{X}} i_{*} O_{D} ≅ i_{*} (L ∣_{D})$ constructed in the proof. The twist $L (- D)$ is again invertible.

Facts & Assumptions

Given: a scheme $X$ , an effective Cartier divisor $i : D ↪ X$ with ideal sheaf $I_{D}$ , and an invertible $O_{X}$ -module $L$ .

[F1]

The divisor $i$ is a closed immersion with $I_{D} = O_{X} (- D)$ invertible, and $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ is a short exact sequence of $O_{X}$ -modules, the first map being the inclusion of the ideal sheaf and the second the quotient map of the closed immersion (Effective Cartier divisors give a short exact sequence, Invertible sheaf of cartier divisor, Closed immersions of schemes).

[F2]

An $O_{X}$ -module is invertible when it is locally free of rank one; then $X$ is covered by open sets $U$ admitting a generator, equivalently a trivialisation $L ∣_{U} ≅ O_{U}$ , restriction to an open subscheme preserves invertibility, and the tensor product, dual and inverse of invertible sheaves are again invertible (Invertible sheaves, The internal Hom sheaf of two module sheaves).

[F3]

For Cartier divisors $D, E$ there are canonical isomorphisms $O_{X} (D + E) ≅ O_{X} (D) \otimes O_{X} (E)$ and $O_{X} (- D) ≅ O_{X} (D)^{\lor}$ , and $O_{X} (0) = O_{X}$ (Addition of Cartier divisors is tensor product of their sheaves).

[F4]

For an invertible $L$ the evaluation $L^{\lor} \otimes_{O_{X}} L \to O_{X}$ is an isomorphism, so $L^{\lor} \otimes L ≅ O_{X}$ canonically (Dual of a line bundle is its tensor inverse).

[F5]

The tensor product of $O_{X}$ -modules is the sheafification of the objectwise tensor product, is functorial in each variable and compatible with restriction to open subschemes; for an $O_{U}$ -module $M$ the unit map $O_{U} \otimes_{O_{U}} M \to M$ is an isomorphism (Tensor product of sheaves of modules, The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[F6]

A sequence of sheaves of modules is exact when at each term the image equals the kernel, and it is exact if and only if all of its stalk sequences are exact (Exact sequences of sheaves, A sequence of abelian sheaves is exact exactly when it is exact on every stalk).

[F7]

For a morphism $f$ the pullback of an $O_{Y}$ -module is $f^{*} G = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} G$ , the direct image is $(f_{*} F) (V) = F (f^{- 1} V)$ , and these constructions are compatible with restriction: for an open $U \subseteq Y$ one has $f^{*} (G ∣_{U}) = f^{*} G ∣_{f^{- 1} U}$ and $(f_{*} F) ∣_{U} = f_{*} (F ∣_{f^{- 1} U})$ (Pullback of a module along a morphism of ringed spaces, Direct image of a sheaf along a continuous map).

[F8]

Local sheaves and local isomorphisms on an open cover which agree on the overlaps, that is a gluing datum, glue to a sheaf, respectively to an isomorphism of sheaves, uniquely (A gluing datum for sheaves on an open cover, Compatible local sheaves glue uniquely up to unique isomorphism, A sheaf on a topological space).

Proof

1.1F1F2F7given

By [F1] the ideal sheaf $I_{D} = O_{X} (- D)$ is invertible and the sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ is exact, with the maps described there; $L$ is invertible by hypothesis [F2]; write $L (- D) = L \otimes_{O_{X}} O_{X} (- D)$ and $L ∣_{D} = i^{*} L$ .

1.2F2F5

Let $A$ be the set of all pairs $(U, τ)$ with $U \subseteq X$ open and $τ : L ∣_{U} \to O_{U}$ an isomorphism of $O_{U}$ -modules; because $L$ is locally free of rank one [F2], every point of $X$ lies in the first component of a member of $A$ , and $A$ is determined by a formula, so no choice is used in indexing by it; for every member and every $O_{U}$ -module $M$ the unit map $O_{U} \otimes_{O_{U}} M \to M$ is an isomorphism [F5].

2.1F5F6step 1.1

Let $(U, τ) \in A$ . Restriction commutes with tensor products and with the structure sheaf [F5], so $τ \otimes id$ identifies $(L \otimes O_{X} (- D)) ∣_{U}$ with $O_{X} (- D) ∣_{U}$ and $(L \otimes i_{*} O_{D}) ∣_{U}$ with $(i_{*} O_{D}) ∣_{U}$ , and the unit isomorphism identifies $L ∣_{U}$ with the second term $O_{U}$ of the restricted exact sequence; applying $(-) \otimes_{O_{X}} L$ to the exact sequence of step 1.1 therefore produces a sequence on $U$ isomorphic term by term, through these identifications, to the exact sequence of step 1.1 restricted to $U$ , whence $0 \to L (- D) ∣_{U} \to L ∣_{U} \to (L \otimes i_{*} O_{D}) ∣_{U} \to 0$ is exact [F6].

2.2F2F5F7F8step 1.2

The canonical isomorphism $L \otimes i_{*} O_{D} ≅ i_{*} (L ∣_{D})$ . For $(U, τ) \in A$ , restricting $τ$ to $i^{- 1} U$ and using [F7] gives trivialisations $(L ∣_{D}) ∣_{i^{- 1} U} ≅ i^{*} (L ∣_{U}) ≅ O_{i^{- 1} U}$ and $(i_{*} O_{D}) ∣_{U} ≅ i_{*} O_{i^{- 1} U}$ ; define $φ_{(U, τ)}$ as the composite of $τ \otimes id$ , the unit isomorphism and the direct image of the inverse trivialisation, an isomorphism $(L \otimes i_{*} O_{D}) ∣_{U} \to i_{*} (L ∣_{D}) ∣_{U}$ . If $(V, σ)$ is a second member and $τ = v σ$ on $U \cap V$ with $v \in O_{X}^{\times} (U \cap V)$ , the factors $v$ and $v^{- 1}$ cancel, so $φ_{(U, τ)}$ and $φ_{(V, σ)}$ agree on the overlap and, by [F8], glue to a global isomorphism $φ : L \otimes_{O_{X}} i_{*} O_{D} \to i_{*} (L ∣_{D})$ independent of the chosen trivialisations.

2.3F2F3F4step 1.1

Applying $(-) \otimes_{O_{X}} L^{\lor}$ to the exact sequence of step 1.1 and using [F4] and [F3], $L (- D) \otimes L^{\lor} ≅ (L \otimes L^{\lor}) \otimes O_{X} (- D) ≅ O_{X} (- D)$ ; as $L$ and $O_{X} (- D)$ are invertible, so is the tensor product $L (- D)$ [F2].

3.1F6step 2.1step 2.2

Let $α = {id}_{L} \otimes (the inclusion I_{D} \subseteq O_{X})$ and let $β$ be the composite of ${id}_{L} \otimes (the quotient O_{X} \to i_{*} O_{D})$ with $φ$ ; on a member $(U, τ)$ of the cover these maps correspond, under the identifications of step 2.1, to the maps of the exact sequence of step 1.1 restricted to $U$ , so the sequence $0 \to L (- D) \to L \to i_{*} (L ∣_{D}) \to 0$ has exact restriction to every member of the cover; since every point of $X$ lies in such a $U$ , all stalk sequences are exact and the sequence is exact by [F6].

4.1step 2.3step 3.1∎

Hence the $O_{X}$ -modules form the short exact sequence $0 \to L (- D) \to L \to i_{*} (L ∣_{D}) \to 0$ with the maps $α$ and $β$ described, and the twist $L (- D)$ is invertible by step 2.3.

No choice principle is used: the cover is the formula-determined set $A$ of all trivialisations of $L$ on opens, and the isomorphisms glue uniquely. For the zero effective divisor $D = \emptyset$ one has $I_{D} = O_{X}$ , $i_{*} O_{D} = 0$ and $O_{X} (- D) = O_{X}$ , so $L (- D) = L$ and the sequence reads $0 \to L \to L \to 0 \to 0$ ; if $X = \emptyset$ all terms are the zero sheaf and the sequence is exact. Taking $L = O_{X}$ recovers the untwisted sequence of Effective Cartier divisors give a short exact sequence.

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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)