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Picard group of a scheme

Definition

Let $X$ be a scheme. The Picard group $Pic (X)$ is the set of isomorphism classes $[L]$ of invertible $O_{X}$ -modules (Invertible sheaves), with product $[L] [M] : = [L \otimes_{O_{X}} M] .$ Its identity is $[O_{X}]$ , and its inverse operation is $[L]^{- 1} = [L^{\lor}], L^{\lor} = H o m_{O_{X}} (L, O_{X}) .$ This is an abelian group. The source states this construction as the Picard group definition and leaves the group-law verification as an exercise (Vakil, §14.1.G, PDF p. 308); the proof below supplies that verification.

Facts & Assumptions

Given: A scheme $X$ and invertible $O_{X}$ -modules $L, M, N$ .

[F1]

Each invertible sheaf is locally isomorphic to $O_{X}$ ; $O_{X}$ is itself invertible (Invertible sheaves).

[F2]

The tensor-product sheaf is the sheafification of the sectionwise module tensor presheaf (Tensor product of sheaves of modules, Sheafification of a presheaf).

[F3]

A compatible morphism from a presheaf to a sheaf induces a unique sheaf morphism from its sheafification (Sheafification is left adjoint to the inclusion of sheaves into presheaves).

[F4]

Compatible local sections of a sheaf glue uniquely (A sheaf on a topological space).

[F5]

Module tensor products have the natural associativity and symmetry isomorphisms $(a \otimes b) \otimes c \mapsto a \otimes (b \otimes c)$ and $a \otimes b \mapsto b \otimes a$ (Symmetry and associativity isomorphisms for tensor products over a commutative ring).

[F6]

The module-tensor unit maps $R \otimes_{R} M \to M$ and $M \otimes_{R} R \to M$ are isomorphisms with inverses $m \mapsto 1 \otimes m$ and $m \mapsto m \otimes 1$ (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[F7]

Tensoring morphisms is functorial and preserves identities and compositions (Module homomorphisms induce tensor-product homomorphisms functorially).

[F8]

The dual of an invertible sheaf is invertible and evaluation gives the isomorphism $L^{\lor} \otimes L ≅ O_{X}$ (Dual of a line bundle is its tensor inverse).

Proof

technique · local trivializations and transition functions

1.1F1F2F6

Choose a common trivializing open cover for $L$ and $M$ , with transition units $g_{i j}$ and $h_{i j}$ . By [F2] and the module tensor-unit isomorphism [F6], $L \otimes M$ is locally $O_{U} \otimes_{O_{U}} O_{U} ≅ O_{U}$ and has transition units $g_{i j} h_{i j}$ ; hence it is invertible.

1.2F2F3F7

If $φ : L \to L^{'}$ and $ψ : M \to M^{'}$ are isomorphisms, the local tensor maps induce a sheaf map by [F2, F3]. Tensoring their inverses gives its inverse by [F7], so the product is well-defined on isomorphism classes.

1.3F1F2F4F5

On a common trivializing cover for $L, M, N$ , the local map $((a \otimes b) \otimes c) \mapsto a \otimes (b \otimes c)$ is the module associator [F5]. It commutes with transition units because $(g_{i j} h_{i j}) k_{i j} = g_{i j} (h_{i j} k_{i j})$ ; the maps and their inverses therefore glue to an associativity isomorphism.

1.4F1F2F4F5

The local map $a \otimes b \mapsto b \otimes a$ from [F5] commutes with transition units because $g_{i j} h_{i j} = h_{i j} g_{i j}$ . It and its reverse-order map glue by [F4] to inverse sheaf maps, giving the commutativity isomorphism.

1.5F1F2F4F6

The local maps $a \otimes b \mapsto a b$ and their inverses $s \mapsto 1 \otimes s$ , $s \mapsto s \otimes 1$ define the left and right unit maps. They commute with transitions because the structure-sheaf transition factor is $1$ , and are the module unit maps [F6]; hence they glue to inverse isomorphisms.

2.1F8step 1.3step 1.4step 1.5

By [F8], evaluation identifies $L^{\lor} \otimes L$ with $O_{X}$ ; the commutativity isomorphism of step 1.4 gives also $L \otimes L^{\lor} ≅ O_{X}$ . Thus every class has the displayed two-sided inverse, and the associativity and unit maps of steps 1.3 and 1.5 make the symmetric product an abelian group.

3.1F1F4∎

If $X = \emptyset$ , the empty-cover sheaf axiom forces the module of sections on its only open set to be the one-element zero module. Hence there is exactly one sheaf of modules, namely $O_{\emptyset}$ ; it is locally free of rank one vacuously, so $Pic (\emptyset)$ is the trivial group.

On a nonempty scheme, the zero sheaf is not locally free of rank one, so it contributes no class. The definition and proof impose no reducedness, Noetherianity, or connectedness assumption on $X$ . They make no additional product-decomposition claim for disconnected schemes.

This item defines only the ordinary group of isomorphism classes. It defines no Picard scheme, representing scheme, or Picard functor; every occurrence of $Pic (X)$ here refers to this group.

Depends on

Used by

Dependency tree · two levels

23 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

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, October 21, 2011 draft, §14.1.G (standard reference, not scraped)