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The Stacks project

Lemma 32.15.1. Let f : X \to Y be a morphism of schemes. Assume f finite type and Y locally Noetherian. Let y \in Y be a point in the closure of the image of f. Then there exists a commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y }

where A is a discrete valuation ring and K is its field of fractions mapping the closed point of \mathop{\mathrm{Spec}}(A) to y. Moreover, we can assume that the image point of \mathop{\mathrm{Spec}}(K) \to X is a generic point \eta of an irreducible component of X and that K = \kappa (\eta ).

Proof. By the non-Noetherian version of this lemma (Morphisms, Lemma 29.6.5) there exists a point x \in X such that f(x) specializes to y. We may replace x by any point specializing to x, hence we may assume that x is a generic point of an irreducible component of X. This produces a ring map \mathcal{O}_{Y, y} \to \kappa (x) (see Schemes, Section 26.13). Let R \subset \kappa (x) be the image. Then R is Noetherian as a quotient of the Noetherian local ring \mathcal{O}_{Y, y}. On the other hand, the extension \kappa (x) is a finitely generated extension of the fraction field of R as f is of finite type. Thus there exists a discrete valuation ring A \subset \kappa (x) with fraction field \kappa (x) dominating R by Algebra, Lemma 10.119.13. Then

\xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[d] \ar[rrr] & & & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \mathop{\mathrm{Spec}}(R) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \ar[r] & Y }

gives the desired diagram. \square


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