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42.16 Preparation for principal divisors

Some of the material in this section partially overlaps with the discussion in Divisors, Section 31.26.

Lemma 42.16.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral.

  1. If $Z \subset X$ is an integral closed subscheme, then the following are equivalent:

    1. $Z$ is a prime divisor,

    2. $Z$ has codimension $1$ in $X$, and

    3. $\dim _\delta (Z) = \dim _\delta (X) - 1$.

  2. If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\dim _\delta (Z) = \dim _\delta (X) - 1$.

Proof. Part (1) follows from the definition of a prime divisor (Divisors, Definition 31.26.2) and the definition of a dimension function (Topology, Definition 5.20.1). Let $\xi \in Z$ be the generic point of an irreducible component $Z$ of an effective Cartier divisor $D \subset X$. Then $\dim (\mathcal{O}_{D, \xi }) = 0$ and $\mathcal{O}_{D, \xi } = \mathcal{O}_{X, \xi }/(f)$ for some nonzerodivisor $f \in \mathcal{O}_{X, \xi }$ (Divisors, Lemma 31.15.2). Then $\dim (\mathcal{O}_{X, \xi }) = 1$ by Algebra, Lemma 10.60.13. Hence $Z$ is as in (1) by Properties, Lemma 28.10.3 and the proof is complete. $\square$

Lemma 42.16.2. Let $f : X \to Y$ be a morphism of schemes. Let $\xi \in Y$ be a point. Assume that

  1. $X$, $Y$ are integral,

  2. $Y$ is locally Noetherian

  3. $f$ is proper, dominant and $R(Y) \subset R(X)$ is finite, and

  4. $\dim (\mathcal{O}_{Y, \xi }) = 1$.

Then there exists an open neighbourhood $V \subset Y$ of $\xi $ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. This lemma is a special case of Varieties, Lemma 33.17.2. Here is a direct argument in this case. By Cohomology of Schemes, Lemma 30.21.2 it suffices to prove that $f^{-1}(\{ \xi \} )$ is finite. We replace $Y$ by an affine open, say $Y = \mathop{\mathrm{Spec}}(R)$. Note that $R$ is Noetherian, as $Y$ is assumed locally Noetherian. Since $f$ is proper it is quasi-compact. Hence we can find a finite affine open covering $X = U_1 \cup \ldots \cup U_ n$ with each $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Note that $R \to A_ i$ is a finite type injective homomorphism of domains such that the induced extension of fraction fields is finite. Thus the lemma follows from Algebra, Lemma 10.113.2. $\square$

Comments (2)

Comment #5092 by Klaus Mattis on

In Lemma 02RM, requirement (3), it should be R(Y) \subset R(X) instead of R(X) \subset R(Y)

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