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    Isomorphic local rings at points give isomorphic open neighborhoods

    P-AGH47LOCRINGISO

    Area
    Algebraic Geometry
    Topics
    Birational Geometry, Local Rings, Varieties
    Status
    draft

    Let \(X\) and \(Y\) be two varieties. Suppose there are points \(P \in X\) and \(Q \in Y\) such that the local rings \(\mco_{P, X}\) and \(\mco_{Q, Y}\) are isomorphic as \(k\)-algebras. Show that there are open sets \(P \in U \subseteq X\) and \(Q \in V \subseteq Y\) and an isomorphism of \(U\) with \(V\) carrying \(P\) to \(Q\).

    Solution

    Fix a \(k\)-algebra isomorphism \[ \alpha:\mco_{P,X}\xrightarrow{\sim}\mco_{Q,Y} \] and write \(\beta=\alpha^{-1}\).

    1. After replacing \(X\) and \(Y\) by affine open neighborhoods of \(P\) and \(Q\), the map \(\alpha\) is induced by a morphism\[ f:V_0\to X \]from an open neighborhood \(V_0\subseteq Y\) of \(Q\).

    Choose affine embeddings \[ X\subseteq\AA^r, \qquad Y\subseteq\AA^s, \] and let \(x_1,\ldots,x_r\) be the coordinate functions on \(X\). For each \(i\), the germ \[ \alpha(x_i)\in\mco_{Q,Y} \] has a regular representative \(g_i\) on some open neighborhood \(V_i\) of \(Q\). On the common neighborhood \[ V'=V_1\cap\cdots\cap V_r \] the tuple \[ (g_1,\ldots,g_r):V'\to\AA^r \] is a morphism.

    Let \(F_1,\ldots,F_m\) generate the ideal \(I(X)\). For each \(j\), the germ of \[ F_j(g_1,\ldots,g_r) \] at \(Q\) equals \[ \alpha(F_j(x_1,\ldots,x_r))=0. \] Hence, after shrinking \(V'\) around \(Q\), all these finitely many functions vanish identically. The tuple therefore lands in \(X\) and defines a morphism \[ f:V_0\to X. \] For every global coordinate function \(x_i\) on \(X\), the germ at \(Q\) of \(x_i\circ f\) is \(\alpha(x_i)\) by construction.

    2. The morphism \(f\) sends \(Q\) to \(P\).

    An isomorphism of local rings sends the unique maximal ideal to the unique maximal ideal, so \[ \alpha(\mathfrak m_{P,X})=\mathfrak m_{Q,Y}. \] For every coordinate function \(x_i\), \[ x_i-x_i(P)\in\mathfrak m_{P,X}. \] Therefore the germ \[ g_i-x_i(P)=\alpha(x_i-x_i(P)) \] lies in \(\mathfrak m_{Q,Y}\) and vanishes at \(Q\). Thus \[ g_i(Q)=x_i(P) \] for all \(i\), which means \(f(Q)=P\).

    Now the induced local homomorphism \[ f_Q^*: \mco_{P,X}\longrightarrow\mco_{Q,Y} \] is defined. It agrees with \(\alpha\) on the image of \(A(X)\). Every element of \(\mco_{P,X}=A(X)_{\mathfrak m_P}\) is a fraction \(a/s\) with \(s\notin\mathfrak m_P\), and both maps send it to the quotient of their values on \(a\) and \(s\). Hence \[ f_Q^*=\alpha. \]

    3. Applying the same construction to \(\beta\) gives an open neighborhood \(U_0\subseteq X\) of \(P\) and a morphism\[ g:U_0\to Y \]with\[ g(P)=Q, \qquad g_P^*=\beta. \]

    Repeat steps <1>1–<1>2 with \(X,P,\alpha\) and \(Y,Q,\beta\) interchanged.

    4. After shrinking around \(P\) and \(Q\), the compositions \(f\circ g\) and \(g\circ f\) are identity morphisms.

    The composition \(f\circ g\) is defined on the open neighborhood \[ U_c=U_0\cap g^{-1}(V_0) \] of \(P\), and \(g\circ f\) is defined on the open neighborhood \[ V_c=V_0\cap f^{-1}(U_0) \] of \(Q\).

    On the local ring at \(P\), \[ (f\circ g)_P^* =g_P^*\circ f_Q^* =\beta\circ\alpha =\operatorname{id}_{\mco_{P,X}}. \] Choose affine coordinate generators \(x_1,\ldots,x_r\) for \(X\). For each \(i\), the two regular functions \[ x_i\circ f\circ g \qquad\text{and}\qquad x_i \] have the same germ at \(P\). After intersecting finitely many neighborhoods, they agree on one open neighborhood \(U_1\subseteq U_c\) of \(P\). Since the coordinate functions determine a map into the affine variety \(X\), this gives \[ f\circ g=\operatorname{id}_{U_1}. \]

    The same argument at \(Q\), using \[ (g\circ f)_Q^* =f_Q^*\circ g_P^* =\alpha\circ\beta =\operatorname{id}_{\mco_{Q,Y}}, \] gives an open neighborhood \(V_1\subseteq V_c\) of \(Q\) on which \[ g\circ f=\operatorname{id}_{V_1}. \]

    5. There are open neighborhoods \(P\in U\subseteq X\) and \(Q\in V\subseteq Y\) such that\[ g:U\xrightarrow{\sim}V \]is an isomorphism carrying \(P\) to \(Q\).

    Set \[ U=U_1\cap g^{-1}(V_1) \] and \[ V=V_1\cap f^{-1}(U). \] Both are open neighborhoods of \(P\) and \(Q\), respectively.

    If \(x\in U\), then \(g(x)\in V_1\) and \[ f(g(x))=x\in U, \] so \(g(x)\in V\). Thus \(g(U)\subseteq V\).

    Conversely, if \(y\in V\), then \(f(y)\in U\) and, because \(y\in V_1\), \[ g(f(y))=y. \] Hence \(y\in g(U)\). Therefore \(g(U)=V\).

    On \(U\) and \(V\), the identities from step 4 show \[ f\circ g=\operatorname{id}_U, \qquad g\circ f=\operatorname{id}_V. \] So \(g:U\to V\) is an isomorphism with inverse \(f|_V\). Step 3 gives \(g(P)=Q\).

    6. Q.E.D.

    Step 5 is precisely the required neighborhood isomorphism.

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