Let \(X\) be any variety and let \(P \in X\). Show that there is a one-to-one correspondence between the prime ideals of the local ring \(\mco_P\) and the closed subvarieties of \(X\) containing \(P\).
Solution
Choose an affine open neighborhood \(U\subseteq X\) of \(P\), and write \[ A=A(U), \qquad \mathfrak m=I_U(P). \] Then \[ \mco_P\cong A_{\mathfrak m}. \]
1. Closed subvarieties of \(X\) containing \(P\) correspond bijectively to closed subvarieties of \(U\) containing \(P\).
If \(Z\subseteq X\) is a closed subvariety containing \(P\), then \(Z\cap U\) is a nonempty open subset of the irreducible space \(Z\), hence is irreducible; it is also closed in \(U\) and contains \(P\).
Conversely, let \(W\subseteq U\) be a closed subvariety containing \(P\). Its closure \(\overline{W}^{X}\) in \(X\) is irreducible and closed, hence is a closed subvariety of \(X\) containing \(P\). Because \(U\) is open in \(X\), \[ \overline{W}^{X}\cap U=\overline{W}^{U}=W, \] the last equality using that \(W\) is closed in \(U\). Likewise, for a closed subvariety \(Z\subseteq X\), the nonempty open subset \(Z\cap U\) is dense in \(Z\), so \[ \overline{Z\cap U}^{X}=Z. \] Thus intersection with \(U\) and closure in \(X\) are inverse operations.
2. Closed subvarieties of \(U\) containing \(P\) correspond bijectively to prime ideals \(\mathfrak p\subseteq A\) contained in \(\mathfrak m\).
Since \(U\) is affine, irreducible closed subsets of \(U\) are exactly the sets \[ Z_U(\mathfrak p) \] for prime ideals \(\mathfrak p\subseteq A\), with inverse given by the vanishing ideal. The point \(P\) belongs to \(Z_U(\mathfrak p)\) exactly when every element of \(\mathfrak p\) vanishes at \(P\), equivalently \[ \mathfrak p\subseteq\mathfrak m. \] This gives the claimed restricted correspondence.
3. Prime ideals of \(A\) contained in \(\mathfrak m\) correspond bijectively to prime ideals of \(A_{\mathfrak m}\).
This is the prime correspondence for localization. Explicitly, \[ \mathfrak p\longmapsto \mathfrak p A_{\mathfrak m} \] sends every prime \(\mathfrak p\subseteq\mathfrak m\) to a prime of \(A_{\mathfrak m}\), while contraction \[ \mathfrak q\longmapsto \mathfrak q\cap A \] sends a prime \(\mathfrak q\subseteq A_{\mathfrak m}\) to a prime ideal of \(A\) disjoint from \(A\setminus\mathfrak m\), hence contained in \(\mathfrak m\). Extension and contraction are inverse under these conditions.
4. Q.E.D.
Using \(\mco_P\cong A_{\mathfrak m}\), compose the bijections of steps <1>1–<1>3. Thus a prime \(\mathfrak q\subseteq\mco_P\) corresponds to the closure in \(X\) of \[ Z_U(\mathfrak q\cap A), \] and every closed subvariety of \(X\) through \(P\) arises uniquely in this way.