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  "body": "\n## Summary\n\nThe fiber product $X \\times_S Y$ of two [[morphisms of schemes|JXRKH6P1Z4]] $X \\to S$ and $Y \\to S$ is their [[pullback|Q05K04QJ71]] in the category of schemes.\nIt always exists, and for affine schemes it is the spectrum of a tensor product of rings.\nFiber products give products of varieties, fibers of morphisms, base change to another field, and the diagonal used to define separatedness.\n\n## Explanations\n\n### L1 human\n\nFor sets, the fiber product of $f \\colon X \\to S$ and $g \\colon Y \\to S$ is the set of pairs $(x, y)$ with $f(x) = g(y)$.\nFor schemes one cannot simply take pairs of points, because points of schemes carry more structure.\nInstead the fiber product is defined by what maps into it look like.\nIn the affine case it is concrete: $\\operatorname{Spec} B \\times_{\\operatorname{Spec} A} \\operatorname{Spec} C = \\operatorname{Spec}(B \\otimes_A C)$.\nFor example, $\\mathbb{A}^1_k \\times_k \\mathbb{A}^1_k = \\operatorname{Spec}(k[x] \\otimes_k k[y]) = \\operatorname{Spec} k[x, y]$ is the plane.\n",
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