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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].\nThe *Picard group* $\\operatorname{Pic}(X)$ is the set of isomorphism classes of [[invertible sheaves|7JCF2TE3V5]] on $X$, with the operation\n$$[\\mathcal{L}] \\cdot [\\mathcal{M}] = [\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{M}].$$\nIt is an [[abelian group|082GHNPJ53]] with identity $[\\mathcal{O}_X]$ and inverse $[\\mathcal{L}]^{-1} = [\\mathcal{L}^\\vee]$, where $\\mathcal{L}^\\vee = \\mathcal{H}om(\\mathcal{L}, \\mathcal{O}_X)$.\n\n## Remarks\n\nFor a field $k$ and $r \\ge 1$, $\\operatorname{Pic}(\\mathbb{P}^r_k) \\cong \\Z$, with $n \\mapsto [\\mathcal{O}(n)]$.\nFor $X = \\operatorname{Spec} A$, $\\operatorname{Pic}(X)$ is the [[Picard group of the ring|DZW58R6T2Q]] $A$.\nIf $A$ is a principal ideal domain, every invertible sheaf on $\\operatorname{Spec} A$ is trivial, so $\\operatorname{Pic}(\\operatorname{Spec} A) = 0$.\nThere is a natural isomorphism $\\operatorname{Pic}(X) \\cong H^1(X, \\mathcal{O}_X^\\times)$ with the first cohomology of the sheaf of units.\n",
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