---- Let $X$ and $Y$ be [[scheme|schemes]]. > [!definition] Definition. ([[proper scheme morphism]]) > A [[morphism of locally ringed spaces|scheme morphism]] $f:X \to Y$ is **proper** if it is [[separated scheme morphism|separated]] [[scheme morphism locally of finite type|of finite type]] and *universally closed*, where by the latter we mean that for any [[morphism of locally ringed spaces|morphism]] $Y' \to Y$, the [[fiber products exist in the category of schemes|induced]] [[categorical pullback|projection morphism]] $X \times_{Y} Y' \to Y'$ is a [[closed scheme morphism|closed morphism]]. ^definition > [!intuition] > Properness is an algebro-geometric notion which captures "[[Hausdorff space|Hausdorff]] + [[compact]]" (or just '[[compact]]' if one is using 'compact' to mean 'Hausdorff + quasicompact'). ^intuition > [!basicnonexample] > Take the inclusion $k \to k[X]$ [[the category of affine schemes is dual to that of rings|inducing]] $\underbrace{ \text{Spec }k[X] }_{ =\mathbb{A}^{1}_{k}=: X } \to \underbrace{ \text{Spec } k }_{ =: Y }$. This morphism is certainly [[closed scheme morphism|closed]]. But it is not proper (which aligns with [[proper map|our topological intuition]]). Indeed, take $Y':=\mathbb{A}^{1}_{k}=\text{Spec }k[T]$. In this case we know the [[categorical pullback|fiber product]] $X \times_{Y} Y'$ is [[spec functor|spec]] of the [[tensor product of algebras|tensor product]] of $k$-[[algebra|algebras]] $k[X] \otimes_{k}k[T]=k[X,T]$: $X \times _{Y} Y'=\text{Spec }k[X,T]=\mathbb{A}^{2}_{k}$ with $p:X \times _Y Y' \to Y'$ induced by the inclusion $k[T] \to k[X,T]$. $p$ is not closed, e.g. because the closed set $Z=V(XT-1)$ (hyperbola) gets sent to $p(Z)=D(T)$, which is not closed. > This is about the most one can easily do just with the definition. In practice, one always checks properness using the [[valuative criterion for properness]]. ^nonexample > [!basicexample] > Let $S$ be a [[graded ring]], $S=\bigoplus_{d \geq 0} S_{d}$. Fact: there is a morphism $\text{Proj }S\to \text{Spec }S_{0}$ which is proper. For example, $\underbrace{ \text{Proj }k[X_{0},\dots,X_{n}] }_{ \mathbb{P}^{n}_{k} } \to \text{Spec }k$ is proper. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```