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  "body": "\n## Summary\n\nThe Picard group $\\operatorname{Pic}(R)$ of a [[commutative ring|0310REJJY8]] $R$ collects the isomorphism classes of rank one [[projective modules|BFFA03GE3V]] over $R$.\nIts group law is the [[tensor product|F0HQXQ0HCD]].\nIt measures how far such modules are from being free.\nFor an [[integral domain|0QVD7WBFGN]] it is the group of [[invertible ideals|E190TAV2S6]] modulo principal ones, and for a [[Dedekind domain|7NPY2CN5KB]] it is the [[ideal class group|WQ46BD3ARY]].\nIt is the Picard group of the affine scheme $\\operatorname{Spec} R$.\n\n## Explanations\n\n### L1 human\n\nA rank one projective module looks like a copy of the ring near every point, but it may be twisted globally.\nThe Picard group collects all these twisted copies, and tensoring two of them gives a third one.\nThe ring itself is the untwisted copy and plays the role of the identity.\nOver $\\Z[\\sqrt{-5}]$, the ideal $(2, 1 + \\sqrt{-5})$ is such a twisted copy, and the Picard group has two elements.\n",
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