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The Stacks project

Lemma 33.42.2. Let X \to S be a morphism of schemes. Let x \in X be a point with image s \in S. If

  1. \mathcal{O}_{X, x} = \mathcal{O}_{S, s},

  2. X is reduced,

  3. X \to S is of finite type, and

  4. S has finitely many irreducible components,

then there exists an open neighbourhood U of x such that f|_ U is an open immersion.

Proof. We may remove the (finitely many) irreducible components of S which do not contain s. We may replace S by an affine open neighbourhood of s. We may replace X by an affine open neighbourhood of x. Say S = \mathop{\mathrm{Spec}}(A) and X = \mathop{\mathrm{Spec}}(B). Let \mathfrak q \subset B, resp. \mathfrak p \subset A be the prime ideal corresponding to x, resp. s. As A is a reduced and all of the minimal primes of A are contained in \mathfrak p we see that A \subset A_\mathfrak p. As X \to S is of finite type, B is of finite type over A. Let b_1, \ldots , b_ n \in B be elements which generate B over A Since A_\mathfrak p = B_\mathfrak q we can find f \in A, f \not\in \mathfrak p and a_ i \in A such that b_ i and a_ i/f have the same image in B_\mathfrak q. Thus we can find g \in B, g \not\in \mathfrak q such that g(fb_ i - a_ i) = 0 in B. It follows that the image of A_ f \to B_{fg} contains the images of b_1, \ldots , b_ n, in particular also the image of g. Choose n \geq 0 and f' \in A such that f'/f^ n maps to the image of g in B_{fg}. Since A_\mathfrak p = B_\mathfrak q we see that f' \not\in \mathfrak p. We conclude that A_{ff'} \to B_{fg} is surjective. Finally, as A_{ff'} \subset A_\mathfrak p = B_\mathfrak q (see above) the map A_{ff'} \to B_{fg} is injective, hence an isomorphism. \square


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