---- > [!definition] Definition. ([[Zariski topology on a ring spectrum]]) > Let $A$ be a [[commutative ring|commutative]] [[ring]]. The **Zariski topology** on the [[prime ideal|spectrum]] $\text{Spec }A$ of $A$ is defined to have as [[closed set|closed sets]] $V(I)=\{ \mathfrak{p} \in \text{Spec }A : \mathfrak{p} \supset I \}$ > where $I$ is an [[ideal]]. The sets $D(f):= \{ \mathfrak{p} \in \text{Spec }A : f \notin \mathfrak{p}\},$ with $f$ ranging over elements of $A$, [[topology generated by a basis|form]] a [[basis for a topology|basis]] for this [[topological space|topology]], and are called **principal open subsets**. Note that $f$ is [[nilpotent element of a ring|nilpotent]] if and only if $D(f)=\emptyset$, as a consequence of [[nilradical equals intersection of all prime ideals]]. There is a [[covariant functor]] $\text{Spec}:\mathsf{CRing}^{\text{op}} \to \mathsf{Aff}$[^1] which takes $A$ to $\text{Spec }A$ and $A \xrightarrow{\varphi}B$ > [!basicproperties] > - $V(I_{1}) \cup \dots \cup V(I_{r})=V(I_{1} \cap \dots \cap I_{r})$ > - $\bigcap_{\alpha}^{}V(I_{\alpha})=V\left( \sum_{\alpha}I_{\alpha} \right)$ > - $V(I)=V(I^{N})$ for any $N \in \mathbb{N}$. Similarly, $V(I)=V(\sqrt{ I })$. > - In fact, the converse holds: $V(I)=V(J) \iff \sqrt{ I }=\sqrt{ J }$. > - $D(f) \cap D(g)=D(fg)$ > - **Functoriality.** The map $\text{Spec }B \xrightarrow{\varphi ^{-1}(\cdot)} \text{Spec } A$ [[the category of affine schemes is dual to that of rings|induced by]] a [[ring homomorphism|ring map]] $A \xrightarrow{\varphi} B$ satisfies $\varphi ^{-1}\big( D(f) \big)=D\big( \varphi(f) \big)$. > - $D(f) = \emptyset$ iff $f$ is [[nilpotent element of a ring|nilpotent]] > - For $f \in A$ the [[localization|localization map]] $A \to A_{f}$ induces a [[homeomorphism]] $D(f) \cong \text{Spec }A_{f}$. For more, see [[distinguished open sets are affine subschemes]]. > - For $\mathfrak{p} \subset R$ prime: $\overline{\{ \mathfrak{p} \}}=V(\mathfrak{p})$. > Eventually will want to show all of these: https://stacks.math.columbia.edu/tag/00E0 ^properties If $\mathfrak{q} \in \overline{\{ \mathfrak{p} \}}$, then $\mathfrak{q}$ belongs to every [[closed set]] that $\mathfrak{p}$ belongs to. In particular, $\mathfrak{q} \in V(\mathfrak{p}) \ni \mathfrak{p}$. For the reverse inclusion: if $\mathfrak{q} \in V(\mathfrak{p})$, and $V(I)$ is a closed set containing $\mathfrak{p}$, then $I \subset \mathfrak{p}$, and $\mathfrak{q} \supset \mathfrak{p} \supset I$ so that $\mathfrak{q} \in V(I)$. So certainly $\overline{\{ \mathfrak{p} \}}=V(\mathfrak{p})$. **$V(I)=V(J) \implies \sqrt{ I }=\sqrt{ J }$.** Suppose $V(I)=V(J)$. [[nilradical equals intersection of all prime ideals|Recalling that]] $\sqrt{ I }=\bigcap_{\mathfrak{p} \supset I } \mathfrak{p},$this is clear. (Other implication is proved elsewhere.) ^definition > [!justification] > To show that the sets $D(f)$ are indeed a basis for the Zariski topology, we will use the [[condition for obtaining a basis from a topology]]. Note that the $D(f)$ are indeed open, since the complement $\text{Spec }A-D(f)=\{ \mathfrak{p} \in \text{Spec }A : \mathfrak{p} \ni f\}=\{ \mathfrak{p} \in \text{Spec }A : \mathfrak{p} \supset \langle f \rangle \}=V(\langle f \rangle)$ is [[closed set|closed]]. > > Now let $U(I)=\text{Spec }A-V(I)$, $U(I)=\{ \mathfrak{p} \in \text{Spec }A : \mathfrak{p} \not \supset I \},$ > be an arbitrary Zariski-open set. Let $\mathfrak{a} \in U$ be arbitrary. We want to find $f \in A$ such that $\mathfrak{a} \in D(f) \subset U(I)$. $\mathfrak{a} \not \supset I$; let $f$ belong to $I$, but not to $\mathfrak{a}$. Then an element of $D(f)=\{ \mathfrak{p} \in \text{Spec }A: f \notin \mathfrak{p} \},$ > a [[prime ideal]] not containing $f$, must not contain $I$ either, since $f \in I$. $\mathfrak{a} \in D(f)$, however, since $f \notin \mathfrak{a}$. > > ---- #### [^1]: We could also call it a [[contravariant functor]] $\text{Spec}:\mathsf{CRing} \to \mathsf{Aff}$. fiber bundles and physical modeling?? opinion dynamics and molecules?? ---- #### 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 > ```