Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 21 / 4

Put $R={\mathcal{O}}_{X,x}$, $\mathfrak{m}={\mathfrak{m}}_x$ and $d=\dim R$. Smoothness makes $R$ a regular local ring, with residue field $k$. A list of local parameters $t_1,\dots ,t_d$ is a regular system of parameters: its images form a basis of the cotangent space of a local ring $\mathfrak{m}/{\mathfrak{m}}^2$. Set $J=(t_1,\dots ,t_d)$. The basis condition says $\mathfrak{m}=J+{\mathfrak{m}}^2$, so the finitely generated module $\mathfrak{m}/J$ equals $\mathfrak{m}(\mathfrak{m}/J)$. The Nakayama lemma gives

We now construct the requested formal power series, rather than assert that a local function has a convergent expansion. The associated graded ring of a regular local ring isIndeed these initial forms generate the graded ring, since the $t_i$ generate $\mathfrak{m}$. The Dimension theorem for Noetherian local rings gives $\dim {\mathrm{gr}}_{\mathfrak{m}}R=d$. A nonzero kernel in the surjection from the polynomial ring would make its quotient have dimension at most $d-1$, so that kernel is zero.

For $f\in R$, choose $P_0\in k$ to be its residue at $x$. Inductively suppose homogeneous polynomials $P_0,\dots ,P_{n-1}$ have been chosen such thatThe graded-ring isomorphism gives a unique homogeneous polynomial $P_n(T_1,\dots ,T_d)$ of degree $n$ representing the class of $r_n$ in ${\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$. Subtracting $P_n(t)$ makes the next remainder lie in ${\mathfrak{m}}^{n+1}$. HenceThis equality is interpreted in the adic completion of a module $\hat{R}=\underleftarrow{\lim }R/{\mathfrak{m}}^n$. The same induction applies to every compatible sequence of residues in this inverse limit, so evaluation $T_i\mapsto t_i$ defines an isomorphism $k[[T_1,\dots ,T_d]]\cong \hat{R}$. It is injective because the lowest nonzero homogeneous term of a series has nonzero initial form. The Krull intersection theorem gives ${\bigcap }_n{\mathfrak{m}}^n=0$, so $R$ embeds in its completion and the series determines $f$. This is the formal expansion at a smooth point; it depends on the chosen regular system of parameters, and does not claim that the infinite sum converges inside $R$. If $d=0$, the local ring is just $k$, and only the constant term is needed.

Choose affine neighborhoods $U_0=\mathrm{Spec}A$ of $x$ and $V_0=\mathrm{Spec}B$ of $y$, and orient the given local ring isomorphism asBoth algebras are finitely presented over $k$. Choose finitely many generators of $B$. Their images under $\alpha$ are fractions in $A$ whose denominators do not vanish at $x$. After inverting the product of those denominators, they define regular functions on an affine open neighborhood of $x$. The finitely many defining relations of $B$ vanish as germs at $x$; each vanishes on a further open neighborhood. Shrinking finitely many times therefore turns those fractions into an actual algebra homomorphism $B\to \mathcal{O}(U_1)$, and hence a morphism of algebraic varieties $F:U_1\to V_0$ inducing $\alpha$ on the local rings.

A local-ring isomorphism preserves the maximal ideal. Its induced residue-field map is the identity of $k$, because it is a $k$-algebra map. Evaluating every coordinate generator consequently shows $F(x)=y$. Apply the identical finite-generator and finite-relation construction to ${\alpha }^{-1}$ to obtain $G:V_1\to U_0$ with $G(y)=x$.

On $U_2=U_1\cap F^{-1}(V_1)$ the composition $GF$ is defined, and on $V_2=V_1\cap G^{-1}(U_1)$ the composition $FG$ is defined. Their maps on local rings are the identities. Thus the coordinate differences between $GF$ and the identity vanish as germs at $x$; finitely many coordinate generators allow us to shrink to $U_3\subseteq U_2$ on which $GF$ actually is the identity. Likewise shrink to $V_3\subseteq V_2$ on which $FG$ is the identity. Finally takeThese are open neighborhoods of the specified points. If $u\in U$, then $F(u)\in V_3$ and $G(F(u))=u\in U_3$, so $F(u)\in V$. Conversely, if $v\in V$, then $G(v)\in U_3$ and $F(G(v))=v\in V_3$, so $G(v)\in U$. Therefore the restrictions are mutual inverse regular maps, provingThis proves isomorphic local rings determine isomorphic neighborhoods using actual regular functions and finite presentations. The conclusion concerns neighborhood isomorphisms, rather than just isomorphisms of formal completions.