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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]], and let $X \\times_Y X$ be the [[fiber product|1DZMT4S9FB]] of $f$ with itself, with projections $p_1, p_2$.\nThe *diagonal* of $f$ is the unique morphism $\\Delta \\colon X \\to X \\times_Y X$ with $p_1 \\circ \\Delta = p_2 \\circ \\Delta = \\id_X$.\nThe morphism $f$ is *separated* if $\\Delta$ is a [[closed immersion|WWRWYSJDET]].\nA scheme $X$ is *separated* if the unique morphism $X \\to \\operatorname{Spec} \\Z$ is separated.\n\n## Remarks\n\nExample: every morphism $\\operatorname{Spec} A \\to \\operatorname{Spec} B$ is separated.\nThe diagonal is $\\operatorname{Spec}$ of the multiplication map $A \\otimes_B A \\to A$, $a \\otimes a' \\mapsto a a'$, which is surjective, so the diagonal is a closed immersion.\nNon-example: let $X$ be the affine line over a field $k$ with a doubled origin, the [[gluing|ABVBHCA9VW]] of two copies of $\\mathbb{A}^1_k$ along the identity of $\\mathbb{A}^1_k \\setminus \\set{0}$.\nThen $X \\to \\operatorname{Spec} k$ is not separated.\nLet $j_1, j_2 \\colon \\mathbb{A}^1_k \\to X$ be the two inclusions, and let $(j_1, j_2) \\colon \\mathbb{A}^1_k \\to X \\times_k X$.\nThe preimage of the diagonal under $(j_1, j_2)$ is $\\mathbb{A}^1_k \\setminus \\set{0}$, the set where $j_1$ and $j_2$ agree.\nThis set is not closed in $\\mathbb{A}^1_k$, so the image of the diagonal is not closed.\nA morphism is separated if and only if its diagonal has closed image, since the diagonal is always an immersion.\n",
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