----- - [[prime ideal|spectrum of a ring]] - [[Zariski topology on a ring spectrum]] - [[structure sheaf on a ring spectrum]] > [!proposition] Proposition. ([[distinguished open sets are affine subschemes]]) > Let $A$ be a ([[commutative ring|commutative]]) [[ring]] and $(\text{Spec } A, \mathcal{O}_{\text{Spec }A})$ the corresponding [[affine scheme]]. Recall that the [[topological space]] $\text{Spec }A$ has a [[basis for a topology|basis]] [[topology generated by a basis|of]] distinguished open sets $D(f)=\{ \mathfrak{p} \in \text{Spec }A: \mathfrak{p} \not \ni f \} \text{ for } f \in A.$ > Then for all $f \in A$, applying the [[spec functor]] to the [[localization|localization map]] $A \xrightarrow{\ell} A_{f}$ produces a [[morphism of locally ringed spaces|scheme isomorphism]] $\big( D(f), \mathcal{O}_{\text{Spec } A} |_{D(f)} \big) \cong_{\ell^{*}} (\text{Spec } A_{f}, \mathcal{O}_{\text{Spec } A_{f}}).$ > Consequently, any [[scheme]] has a basis of principal open affines. > > [!proof]- Proof. ([[distinguished open sets are affine subschemes]]) > ~ Recall that $\ell^{*}=\text{Spec } \ell$ is the [[contraction of an ideal|contraction map]] $\ell^{-1}(\cdot)$. Per [[extension and contraction under localization]], there is a [[bijection]] $\{ \mathfrak{p} \in \text{Spec } A : \mathfrak{p} \not \ni f \} \leftrightarrow \text{Spec }A_{f}$ given by [[extension of an ideal|extension]] and [[contraction of an ideal|contraction]] along the [[localization|localization map]] $A \to A_{f}$. Both directions are [[continuous]], so it is in fact a [[homeomorphism]]. Thus, $\ell^{*}$ witnesses $D(f) \cong \text{Spec }A_{f}$ as [[topological space|topological spaces]]. It remains to be shown that $(\ell^{*})^{\sharp}:\mathcal{O}_{\text{Spec }A |_{D(f)}} \to \ell^{*}_{*}\mathcal{O}_{\text{Spec }A_{f}}$ is a [[morphism of (pre)sheaves|sheaf isomorphism]], where $\ell^{*}_{*}\mathcal{O}_{\text{Spec }A_{f}}$ denotes the [[differential of a smooth map between smooth manifolds|pushforward]] of $\text{Spec }A_{f}$ along the contraction map $\ell^{*}$. [[sheaf isomorphism iff isomorphism on stalks|It is enough to check that]], for all $\mathfrak{p} \in D(f)$, the [[(pre)sheaf stalk|stalk map]] $\begin{align} (\ell^{*})^{\sharp}_{\mathfrak{p}}: (\mathcal{O}_{\text{Spec } A |_{D(f)}})_{\mathfrak{p}} & \to (\ell^{*}_{*} \mathcal{O}_{\text{Spec } A_{f}})_{\mathfrak{p}} \\ [U, s] & \mapsto [U, (\ell^{*})^{\sharp}_{U}(s) ] \end{align}$ is a (local) [[ring isomorphism]]. This follows from the fact that this map is isomorphic to the the map $\begin{align} A_{\mathfrak{p}} & \xrightarrow{\sim} (A_{f})_{\mathfrak{p}A_{\mathfrak{p}}} \\ \frac{a}{g} & \mapsto \frac{\left( \frac{a}{1} \right)}{ \left( \frac{g}{1} \right)} \end{align},$ since the latter is an isomorphism. Indeed, [[extension and contraction under localization]] give rise to an obvious isomorphism $\begin{align} (\ell^{*}_{*}\mathcal{O}_{\text{Spec }A_{f}})_{\mathfrak{p}} &\xrightarrow{\sim}\mathcal{O}_{\text{Spec } A_{f}, \mathfrak{p}A_{\mathfrak{p}}} \\ [U \ni \mathfrak{p}, s \in \mathcal{O}_{\text{Spec } A_{f}}\big( (\ell^{*}) ^{-1}(U) \big)] & \mapsto [(\ell^{*}) ^{-1}(U) \ni \mathfrak{p}A_{\mathfrak{p}}, s] \end{align}$ which, together with the usual isomorphisms of the form $\mathcal{O}_{\text{Spec }A, \mathfrak{p}} \cong A_{\mathfrak{p}}, s \mapsto s(\mathfrak{p})$, produce a commutative square: ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{xcolor} \definecolor{thistle}{RGB}{216,191,216} \begin{document} {\small \begin{tikzcd}[column sep=0.2em] {[U \ni \mathfrak{p}, s: U \to \coprod_{\mathfrak{p} \in U} A_\mathfrak{p}]} \arrow[dddd, maps to] \arrow[rrrr, maps to] & & & & {[U, (\ell^*)^\sharp_U(s)]} \arrow[dd, maps to] \\ & (\mathcal{O}_{\text{Spec A}|_{D(f)}})_\mathfrak{p} \arrow[rr] \arrow[dd, "\sim"'] & & (\ell^*_*\mathcal{O}_{\text{Spec }A_f})_\mathfrak{p} \arrow[d, "\sim"] & \\ & & & (\mathcal{O}_{\text{Spec }A_f})_{\mathfrak{p}A_\mathfrak{p}} \arrow[d, "\sim"] & {[(\ell^*)^{-1}(U), (\ell^*)^\sharp_U(s)]} \arrow[dd, maps to] \\ & A_\mathfrak{p} \arrow[rr, "\sim"'] & & (A_f)_{\mathfrak{p} A_{\mathfrak{p}}} & \\ s(\mathfrak{p}) \arrow[rrrr, maps to, color=thistle] & & & & (\ell^*)^\sharp_U(s) \ (\mathfrak{p}A_\mathfrak{p}) \end{tikzcd} } \end{document} ``` where we can see that the $\text{\textcolor{thistle}{bottom mapping}}$ holds by unpacking the (somewhat involved?) definition of $(\ell^{*})^{\sharp}_{U}(s)$: assuming $s(\mathfrak{p})=\frac{a}{g} \in A_{\mathfrak{p}}$, $\begin{align} (\ell^{*})^{\sharp}_{U}(s) \ (\mathfrak{p}A_{\mathfrak{p}}) &= \ell_{\mathfrak{p}_{A \mathfrak{p}}}\big( s(\underbrace{ \ell^{*}(\mathfrak{p}A_{\mathfrak{p}} }_{ =\mathfrak{p} })) \big) \\ &= \ell_{\mathfrak{p}A\mathfrak{p}}\left( \frac{a}{g} \right) \\ &= \frac{\ell(a)}{\ell(g)} \\ &= \frac{\left( \frac{a}{1} \right)}{\left( \frac{g}{1} \right)}. \end{align}$ ----- #### ---- #### 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 > ```