---- > [!definition] Definition. (open [[subscheme]], open immersion) > Let $X$ be a [[scheme]] with [[scheme|structure sheaf]] $\mathcal{O}_{X}$. An **open subscheme** is a [[scheme]] $U$, whose [[topological space]] is an open subset of $X$, and whose [[scheme|structure sheaf]] $\mathcal{O}_{U}$ is [[sheaf isomorphism iff isomorphism on stalks|isomorphic]] to the [[restriction sheaf]] $\mathcal{O}_{X} |_{U}$. > An **open immersion** $f:X \to Y$ is a [[morphism of locally ringed spaces|morphism of schemes]] which induces an [[isomorphism]] between $X$ and an open subscheme $U$ of $Y$. Unlike in differential geometry, these immersions are always [[injective sheaf morphism|injective]]. > > > > [!justification]- > > Need to verify that $(U, \mathcal{O}_{X} |_{U})$ actually *is* a [[scheme]] for any open $U \subset X$. If $\{ U_{i} \}$ is a [[cover|cover]] of $X$ by [[affine scheme|open affines]] $(U_{i}, \mathcal{O}_{X} |_{U_{i}}) \cong ( \text{Spec }A_{i}, \mathcal{O}_{\text{Spec }A_{i}})$, then we can obtain a cover of $U$ by open affines by first considering the open cover $\{ U \cap U_{i} \}$ of $U$, then identify each $U \cap U_{i}$ with an open subset of $\text{Spec }A_{i}$ to cover $U \cap U_{i}$ with [[basis for a topology|basic open sets]] $D(f_{i})$ for $f_{i} \in A_{i}$. Since each $D(f_{i})$ [[distinguished open sets are affine subschemes|is affine]], this gives $U \cap U_{i}$, and in turn $U$, a cover by open affines. > > [!definition] Definition. (closed [[subscheme]]) > A **closed immersion** $f:X \to Y$ is a [[morphism of locally ringed spaces|morphism of schemes]] which is at the [[topological space|topological]] level a [[homeomorphism]] between $X$ and a [[closed set|closed subset]] of $Y$, and at the [[scheme|structure sheaf]] level is a [[surjective sheaf morphism|surjective]] [[morphism of (pre)sheaves|sheaf morphism]] $f^{\sharp}:\mathcal{O}_{Y} \to f_{*}\mathcal{O}_{X}$. > > A **closed subscheme** is an [[equivalence class]] of closed immersions, where we identify $(f: X \to Y) \sim (f': X' \to Y)$ iff there is an [[isomorphism]] $X \to X'$ make the diagram commute: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRQAmclVqMWbdgHJuvEBmx4CRAIyk14+s1aIQATW7iYUAObwioAGYAnCAFskZEDghINE3WwA6P7E48NvZOiC5uSKJeUvrWINQMdABGMAwACvwqQiC2WGYAFjjywY4e1BGIUToxINZyXBRcQA > \begin{tikzcd} > X \arrow[rr, "\sim"] \arrow[rd, "f"'] & & X' \arrow[ld, "f'"] \\ > & Y & > \end{tikzcd} > \end{document} > ``` > > [!basicexample] $D(f)$ as an open subscheme of $\text{Spec }A$. > $\big( D(f), \mathcal{O}_{\text{Spec } A} |_{D(f)} \big) \cong \text{Spec } A_{f}$ is by definition an open subscheme of $\text{Spec }A$. ^basic-example > [!basicexample] $V(I)$ as a closed subscheme of $\text{Spec }A$. > > Consider an [[affine scheme]] $Y=\text{Spec }A$ for $A$ some ([[commutative ring|commutative]]) [[ring]]. Endowing the basic open sets $D(f)$, $f \in A$, with open subscheme structure is easy. Analogously, we would like to view the [[closed set|closed sets]] $V(I)$, $I$ an [[ideal]] of $A$, as closed subschemes $\big( V(I), \mathcal{O}_{V(I)} \big)$. > > This is done as follows. Let $X=\text{Spec } \frac{A}{I}$. [[the correspondence theorem for rings|Recall]] that the [[prime ideal|prime ideals]] of $\frac{A}{I}$ are precisely of the form $\frac{\mathfrak{p}}{I}$ for $I \subset\mathfrak{p} \subset A$ [[prime ideal|prime]]. The [[quotient ring|quotient map]] $A \xrightarrow{\pi} \frac{A}{I}$ [[the category of affine schemes is dual to that of rings|induces an affine scheme morphism]] $\begin{align} > \text{Spec } \frac{A}{I} &\xrightarrow{f:=\pi^{*}} \text{Spec }A \\ > \frac{\mathfrak{p}}{I} & \mapsto \pi ^{-1}\left( \frac{\mathfrak{p}}{I} \right)=\mathfrak{p}. > \end{align}$ > [[the correspondence theorem for rings|It is clear]] that $f$ topologically [[homeomorphism|identifies]] $\text{Spec } \frac{A}{I}$ with $\im f=V(I)$. > > > > At the structure sheaf level, [[spec functor|we know]] that the [[(pre)sheaf stalk|stalk map]] $f^{\sharp}_{\mathfrak{p}}:\mathcal{O}_{\text{Spec } A, f\left( \mathfrak{p} \right)} \to (f_{*} \mathcal{O}_{\text{Spec } \frac{A}{I}, \frac{\mathfrak{p}}{I} })$ is [[isomorphism|isomorphic]] to the [[ring homomorphism]] $\begin{align} > \pi_{\frac{\mathfrak{p}}{I}}: A_{\frac{\mathfrak{p}}{I}}& \to \left( \frac{A}{I} \right)_{\mathfrak{p}} \\ > \frac{a}{s} & \mapsto \frac{\pi(a)}{\pi(s)}. > \end{align}$ This is [[surjection|surjective]], because $\pi$ is. Since we have [[surjection|surjections]] on [[(pre)sheaf stalk|stalks]], [[sheaf morphism injectivity and surjectivity can be tested on stalks|we conclude]] that $f^{\sharp}$ is a [[surjective sheaf morphism]]. Thus, $f$ is a closed immersion $\text{Spec } \frac{A}{I} \to V(I)$. > > Now we take $\mathcal{O}_{V\big(I\big)}:= f_{*}\mathcal{O}_{\text{Spec}(A / I)}$, and the [[equivalence class]] represented by $\big( V(I), \mathcal{O}_{V(I)} \big)$ is a closed subscheme of $\text{Spec }A$. > [!basicnonexample] Exemplifying subtlety in the closed subscheme definition. > In light of the above example $\text{Spec }k[X,Y]/\langle X^{2}, XY \rangle=^{\text{Corr.Thm.}}V(\langle X^{2}, XY \rangle)$ is a closed subscheme of $\mathbb{A}^{2}_{k}=\text{Spec }k[X,Y]$. [[Zariski topology on a ring spectrum|Since]] $V(\langle X^{2}, XY \rangle)=V(\langle X \rangle)$, topologically this space is 'the $Y$-axis', no different than the topological space $\text{Spec }k[Y]$. > But *as (closed sub)schemes* $\text{Spec } k[X,Y] / \langle X^{2}, XY \rangle \neq \text{Spec }k[Y]$. They agree 'away from the origin', in the sense that if we localize at (i.e. can divide by) $Y$ we get the same thing. But at the origin the picture has some fuzz. > ![[Pasted image 20250508145646.png]] ^nonexample ---- #### ---- #### 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 > ```