Picard group

Builds on Ringed space Invertible sheaf Abelian group

Definition

Let $(X, \mathcal{O}_X)$ be a ringed space. The Picard group $\operatorname{Pic}(X)$ is the set of isomorphism classes of invertible sheaves on $X$ , with the operation $[\mathcal{L}] \cdot [\mathcal{M}] = [\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{M}].$ It is an abelian group with identity $[\mathcal{O}_X]$ and inverse $[\mathcal{L}]^{-1} = [\mathcal{L}^\vee]$ , where $\mathcal{L}^\vee = \mathcal{H}om(\mathcal{L}, \mathcal{O}_X)$ .

Remarks

For a field $k$ and $r \ge 1$ , $\operatorname{Pic}(\mathbb{P}^r_k) \cong \Z$ , with $n \mapsto [\mathcal{O}(n)]$ . For $X = \operatorname{Spec} A$ , $\operatorname{Pic}(X)$ is the Picard group of the ring $A$ . If $A$ is a principal ideal domain, every invertible sheaf on $\operatorname{Spec} A$ is trivial, so $\operatorname{Pic}(\operatorname{Spec} A) = 0$ . There is a natural isomorphism $\operatorname{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times)$ with the first cohomology of the sheaf of units.

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Abelian group concept
Invertible sheaf concept
Ringed space concept
Picard group definition

Tag	`JZAKGD68TY`
URI	`m2://JZAKGD68TY`
Kind	definition
Writing	W0 stub
MSC	14C22, 14C20
Defines	Picard group
Uses	Ringed space, Invertible sheaf, Abelian group

Picard group

Definition

Remarks

Reading path

Further reading

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