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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].\nThe morphism $f$ is *proper* if it satisfies the following three conditions.\n\n1. $f$ is [[separated|X30G4V7VAJ]].\n2. $f$ is [[of finite type|AHTE5DVX61]].\n3. $f$ is [[universally closed|W0B0BBAH35]].\n\n## Remarks\n\nLet $k$ be a field.\nThe structure morphism $\\mathbb{P}^n_k \\to \\operatorname{Spec} k$ of projective $n$-space is proper.\nIt is separated and of finite type, and it is universally closed by the main theorem of elimination theory, which can also be derived from the [[valuative criterion|QZN6P7AG33]].\nThe structure morphism $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$ is separated and of finite type but not proper.\nAfter base change along $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$ it becomes the projection $\\mathbb{A}^2_k \\to \\mathbb{A}^1_k$, $(x, y) \\mapsto x$.\nThis projection maps the closed curve $xy = 1$ onto $\\mathbb{A}^1_k \\setminus \\set{0}$, which is not closed, so clause 3 fails.\nClosed immersions and finite morphisms are proper, and proper morphisms are stable under composition and base change.\n",
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