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Tag 02JQ

Chapter 28: Morphisms of Schemes > Section 28.6: Scheme theoretic image

Lemma 28.6.5. Let $f : X \to Y$ be a quasi-compact morphism. Let $Z$ be the scheme theoretic image of $f$. Let $z \in Z$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \mathop{\rm Spec}(A) \ar[r] & Z \ar[r] & Y } $$ such that the closed point of $\mathop{\rm Spec}(A)$ maps to $z$. In particular any point of $Z$ is the specialization of a point of $f(X)$.

Proof. Let $z \in \mathop{\rm Spec}(R) = V \subset Y$ be an affine open neighbourhood of $z$. By Lemma 28.6.3 the intersection $Z \cap V$ is the scheme theoretic image of $f^{-1}(V) \to V$. Hence we may replace $Y$ by $V$ and assume $Y = \mathop{\rm Spec}(R)$ is affine. In this case $X$ is quasi-compact as $f$ is quasi-compact. Say $X = U_1 \cup \ldots \cup U_n$ is a finite affine open covering. Write $U_i = \mathop{\rm Spec}(A_i)$. Let $I = \mathop{\rm Ker}(R \to A_1 \times \ldots \times A_n)$. By Lemma 28.6.3 again we see that $Z$ corresponds to the closed subscheme $\mathop{\rm Spec}(R/I)$ of $Y$. If $\mathfrak p \subset R$ is the prime corresponding to $z$, then we see that $I_{\mathfrak p} \subset R_{\mathfrak p}$ is not an equality. Hence (as localization is exact, see Algebra, Proposition 10.9.12) we see that $R_{\mathfrak p} \to (A_1)_{\mathfrak p} \times \ldots \times (A_1)_{\mathfrak p}$ is not zero. Hence one of the rings $(A_i)_{\mathfrak p}$ is not zero. Hence there exists an $i$ and a prime $\mathfrak q_i \subset A_i$ lying over a prime $\mathfrak p_i \subset \mathfrak p$. By Algebra, Lemma 10.49.2 we can choose a valuation ring $A \subset K = f.f.(A_i/\mathfrak q_i)$ dominating the local ring $R_{\mathfrak p}/\mathfrak p_iR_{\mathfrak p} \subset f.f.(A_i/\mathfrak q_i)$. This gives the desired diagram. Some details omitted. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 867–881 (see updates for more information).

    \begin{lemma}
    \label{lemma-reach-points-scheme-theoretic-image}
    Let $f : X \to Y$ be a quasi-compact morphism.
    Let $Z$ be the scheme theoretic image of $f$.
    Let $z \in Z$. There exists a valuation ring $A$ with
    fraction field $K$ and a commutative diagram
    $$
    \xymatrix{
    \Spec(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\
    \Spec(A) \ar[r] & Z \ar[r] & Y
    }
    $$
    such that the closed point of $\Spec(A)$ maps to $z$. In particular
    any point of $Z$ is the specialization of a point of $f(X)$.
    \end{lemma}
    
    \begin{proof}
    Let $z \in \Spec(R) = V \subset Y$ be an affine open
    neighbourhood of $z$. By
    Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}
    the intersection $Z \cap V$ is the scheme theoretic image of
    $f^{-1}(V) \to V$. Hence we may replace $Y$ by $V$
    and assume $Y = \Spec(R)$ is affine.
    In this case $X$ is quasi-compact as $f$ is quasi-compact.
    Say $X = U_1 \cup \ldots \cup U_n$
    is a finite affine open covering. Write $U_i = \Spec(A_i)$.
    Let $I = \Ker(R \to A_1 \times \ldots \times A_n)$.
    By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}
    again we see that $Z$ corresponds to the closed subscheme
    $\Spec(R/I)$ of $Y$. If $\mathfrak p \subset R$ is
    the prime corresponding to $z$, then we see that
    $I_{\mathfrak p} \subset R_{\mathfrak p}$ is not an
    equality. Hence (as localization is exact, see
    Algebra, Proposition \ref{algebra-proposition-localization-exact})
    we see that
    $R_{\mathfrak p} \to
    (A_1)_{\mathfrak p} \times \ldots \times (A_1)_{\mathfrak p}$
    is not zero. Hence one of the rings $(A_i)_{\mathfrak p}$ is not zero.
    Hence there exists an $i$ and a prime $\mathfrak q_i \subset A_i$
    lying over a prime $\mathfrak p_i \subset \mathfrak p$.
    By Algebra, Lemma \ref{algebra-lemma-dominate} we can choose a valuation ring
    $A \subset K = f.f.(A_i/\mathfrak q_i)$ dominating
    the local ring
    $R_{\mathfrak p}/\mathfrak p_iR_{\mathfrak p} \subset f.f.(A_i/\mathfrak q_i)$.
    This gives the desired diagram. Some details omitted.
    \end{proof}

    Comments (4)

    Comment #2762 by BCnrd on August 10, 2017 a 4:35 pm UTC

    The notation $f.f.$ to denote "fraction field" (twice) near the end of the proof seems kind of awful (not only because in the argument there is a map called $f$). Is this really the standard notation in this document? Why not ${\rm{Frac}}$ instead?

    Comment #2765 by Dario WeiƟmann on August 10, 2017 a 8:58 pm UTC

    Why not use the notation $Q(-)$ introduced in Example 10.9.8 (3)? Or - as in this case we are talking about the residue field at a prime - use the notation $\kappa(-)$ introduced in 25.2.1 (2)?

    Comment #2766 by BCnrd on August 11, 2017 a 12:18 am UTC

    Dario's suggestion of $\kappa(\mathfrak{p})$ for the fraction field of $A/\mathfrak{p}$ seems fine since such notation has been standard from EGA, and I think the meaning of ${\rm{Frac}}$ is also immediately recognized by anyone, but I am not a fan of $Q(\cdot)$ to denote the fraction field of a domain since it is not sufficiently universally known and so potentially a bit obscure (though admittedly anyone who has actually understood the proof up to there can infer what it must mean). In a work this massive, which essentially nobody reads linearly, it is best not to use slightly non-standard notation unless there is an easily-identified and systematically maintained list of all such notation, but alas there seems to be no such list; the Notation section in the "Preliminaries" Part is a bit brief!)

    Comment #2767 by sdf on August 11, 2017 a 5:03 pm UTC

    There is a fairly standard notation of $\mathrm{ff}$ or $\mathrm{FF}$ or $\mathrm{FoF}$ for field of fractions at least in my part of the world. But having a full-stop as part of the notation is aesthetically less than ideal, this has been pointed out before I think.

    There are also 2 comments on Section 28.6: Morphisms of Schemes.

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