---- > [!definition] Definition. ([[scheme over a field]]) > Let $(X, \mathcal{O}_{X})$ be a [[scheme]] and $k$ be a [[field]]. What does a [[morphism of locally ringed spaces|morphism]] $(f, f^{\sharp}):(X, \mathcal{O}_{X})\to(\text{Spec }k, \mathcal{O}_{\text{Spec }k})$ mean? No information in the [[continuous|continuous map]] $f:X \to \text{Spec }k=\{ (0) \}$, but also have the map $f^{\sharp}:\mathcal{O}_{\text{Spec } k} \to f_{*}\mathcal{O}_{X}$ , i.e., a [[ring homomorphism]] [^1] $f^{\sharp}_{\text{Spec } k}:k \hookrightarrow \mathcal{O}_{X}(X)$ endowing the space of global sections $\mathcal{O}_{X}(X)=\Gamma(X, \mathcal{O}_{X})$ with $k$-[[algebra]] structure. > Post-composition with restriction homomorphisms turns $\mathcal{O}_{X}(V)$ into a $k$-[[algebra]] for each open $V \subset X$, yielding a sheaf of $k$-[[algebra|algebras]] (so the [[local ring|local rings]]/[[(pre)sheaf stalk|stalks]] $\mathcal{O}_{X, p}$ are also $k$-[[algebra|algebras]] for each $p \in X$). > We say $X$, together with $f:X \to \text{Spec }k$, is a **scheme defined over the field $k$**, or **$k$-scheme**. ^definition Any $k$-[[algebra]] $k \hookrightarrow A$ [[the category of affine schemes is dual to that of rings|dualizes]] to an [[affine scheme|affine]] $k$-scheme $\text{Spec }A \to \text{Spec }k$. A notable case is when $A=k[T_{1},\dots,T_{n}] / I$, $I=\sqrt{ I }$, is the [[coordinate ring]] of a $k$-[[algebraic set|algebraic subset]] (or [[affine variety]] if we further choose $I$ to be [[prime ideal|prime]]). Recall that if $k$ is [[algebraically closed]], [[Hilbert's geometry-algebra correspondence|we get a]] [[bijection]] $Z(I) \leftrightarrow \text{mSpec }A,$ where $Z(I)=\{ (a_{1},\dots,a_{n}): f(a_{1},\dots,a_{n})=0 \ \fa f \in I \} \subset \mathbb{A}^{n}_{k}$. The scheme-theoretic picture looks more generally at the $k$-scheme $X=\text{Spec }A \supset \text{mSpec }A \leftrightarrow V(I)$. Note that the [[affine scheme]] $X$ has more points than the classical variety $V(I)$! For the simplest possible example, consider $X=\text{Spec }k[T]$, $k=\overline{k}$. The nonzero prime ([[prime iff maximal for nonzero ideals in PID|=]][[maximal ideal|maximal]]) [[ideal|ideals]] of $k[T]$ are $\{\langle T-a \rangle: a \in k\}$, in [[bijection]] with $\mathbb{A}^{1} \cong k$. But then there is also the zero ideal, which is prime but not maximal. The [[Zariski topology on a ring spectrum|Zariski topology]] says the [[closed set|closed sets]] are - $\text{Spec }A$ ($V(\{ 0 \})$) - Finite subsets of $\mathbb{A}^{1} \cong \text{mSpec } A$ Notice that $\overline{\{ 0 \}}$, the intersection of all closed sets containing the zero ideal, equals all of $\text{Spec }A$. So we have a point (a 'generic point') that is [[dense]] in $\text{Spec }A$. We visualize the situation as a copy of $\mathbb{A}^{1} \cong k$ 'with some fuzz at zero': ![[Pasted image 20250507142458.png]] This is the **scheme-theoretic affine line**. *my notation is really bad in light of this new definition* ---- #### [^1]: To be very explicit, what is happening here is that $f^{\sharp}$ is a [[natural transformation]] specified via component maps $f^{\sharp}_{V}$ for $V$ an open subset of the domain. In this case, the domain is $\text{Spec }k$, whose open sets are precisely $\emptyset$ and $\text{Spec } k$. So specifying $f^{\sharp}$ amounts to just specifying $f^{\sharp}_{\text{Spec } k}:\mathcal{O}_{\text{Spec }k}(\text{Spec } k) \to \mathcal{O}_{X}\big( f ^{-1}(\text{Spec } k) \big)$. $\mathcal{O}_{\text{Spec }k}(\text{Spec }k)$, by definition, consists of all maps $(0) \to k$ and is hence naturally [[isomorphism|isomorphic]] to $k$ itself. (This is the second property in [[structure sheaf on a ring spectrum]].) And, of course, $f^{-1}(\text{Spec } k)=X$. So $f^{\sharp}_{\text{Spec }k}$ (and hence $f^{\sharp}$) is really a map $k \to \mathcal{O}_{X}(X)$. Finally, we wrote $\hookrightarrow$ instead of just $\to$ in light of [[every nonzero ring homomorphism out of a field is injective]]. ---- #### 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 > ```