Let \(A\in \Ring\) and \(X\da \Spec(A)\), and for \(f\in A\) let \(D(f) \da V(\gens{f})^c\). Show that there is an isomorphism of ringed spaces \[ (D(f), \ro{\OO_X}{D(f)}) \iso \Spec(A_f) .\]
Hint
- Take \(\iota: A\to A_f\) and the induced map \(\iota^*: \Spec A_f \to \Spec A\).
- Use \(\Spec S^{-1}A \cong \ts{\mfp\in \Spec A \st \mfp \intersect S = \emptyset }\), so \(\Spec A_f = \ts{\mfp\in \Spec A \st \mfp \not\supseteq \gens{f}}\).
- Construct \(\psi \da \iota^*\), and check \(D(g/f^k) \xrightarrow{\psi} D(gf)\) and \(D(g) \xrightarrow{\psi^{-1}} D(g/1)\).
- Use \(\ro{\OO_{\Spec A}}{D(f)}(D(g)) = (A_f)_g\) and define \(\psi^\# = \id\).
Solution
Let \(\iota\colon A\to A_f\), \(a\mapsto a/1\), let \(Y\da\Spec A_f\), and let \(\psi\colon Y\to X\) be \(\psi(\mfq)\da\iota^{-1}(\mfq)\).
1. \(\psi\) is a homeomorphism of \(Y\) onto \(D(f)\).
The primes of \(A_f\) correspond to the primes \(\mfp\) of \(A\) with \(f\notin\mfp\), by \(\mfq\mapsto\iota^{-1}(\mfq)\) with inverse \(\mfp\mapsto\mfp A_f\); so \(\psi\) is a bijection onto \(D(f)\). For \(a\in A\), \(\psi^{-1}(D(a))=D(a/1)\), so \(\psi\) is continuous. Every \(a/f^k\in A_f\) differs from \(a/1\) by a unit, so the sets \(D(a/1)\) form a basis of \(Y\), and \(\psi(D(a/1))=D(a)\cap D(f)=D(af)\) is open. So \(\psi\) is open onto \(D(f)\).
2. For \(a\in A\), there is an isomorphism \(\OO_X(D(af))\to\OO_Y(\psi^{-1}(D(af)))\), compatible with restriction.
By step 1, \(\psi^{-1}(D(af))=D(a/1)\). The sections are \(\OO_X(D(af))=A_{af}\) and \(\OO_Y(D(a/1))=(A_f)_{a/1}\), and \(x/(af)^n\mapsto (x/1)/(af/1)^n\) is an isomorphism \(A_{af}\to(A_f)_{a/1}\), since both rings are \(A\) with \(a\) and \(f\) inverted. For \(D(bf)\subseteq D(af)\) the restriction maps on both sides are the canonical maps between localizations of \(A\), so the isomorphisms commute with them.
3. Q.E.D.
The sets \(D(af)\), \(a\in A\), form a basis of \(D(f)\) closed under intersection. By step 2, \(\psi^\#\colon\ro{\OO_X}{D(f)}\to\psi_*\OO_Y\) is defined and bijective on this basis and commutes with restriction, so it extends uniquely to an isomorphism of sheaves on \(D(f)\). With step 1, \((\psi,\psi^\#)\colon(Y,\OO_Y)\to(D(f),\ro{\OO_X}{D(f)})\) is an isomorphism of ringed spaces, and its inverse is the required isomorphism.