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Tag 0E0N

Chapter 36: More on Morphisms > Section 36.45: Stein factorization

Lemma 36.45.8. Let $f : X \to S$ be a morphism of schemes. Assume

  1. $f$ is proper, flat, and of finite presentation, and
  2. the geometric fibres of $f$ are reduced.

Then the function $n_{X/S} : S \to \mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ is locally constant.

Proof. By Lemma 36.45.7 the function $n_{X/S}$ is lower semincontinuous. For $s \in S$ consider the $\kappa(s)$-algebra $$ A = H^0(X_s, \mathcal{O}_{X_s}) $$ By Varieties, Lemma 32.9.3 and the fact that $X_s$ is geometrically reduced $A$ is finite product of finite separable extensions of $\kappa(s)$. Hence $A \otimes_{\kappa(s)} \kappa(\overline{s})$ is a product of $\beta_0(s) = \dim_{\kappa(s)} H^0(E \otimes^\mathbf{L} \kappa(s))$ copies of $\kappa(\overline{s})$. Thus $X_{\overline{s}}$ has $\beta_0(s) = \dim_{\kappa(s)} A$ connected components. In other words, we have $n_{X/S} = \beta_0$ as functions on $S$. Thus $n_{X/S}$ is upper semicontinuous by Derived Categories of Schemes, Lemma 35.28.1. This finishes the proof. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 13301–13311 (see updates for more information).

    \begin{lemma}
    \label{lemma-proper-flat-geom-red}
    Let $f : X \to S$ be a morphism of schemes. Assume
    \begin{enumerate}
    \item $f$ is proper, flat, and of finite presentation, and
    \item the geometric fibres of $f$ are reduced.
    \end{enumerate}
    Then the function $n_{X/S} : S \to \mathbf{Z}$
    counting the numbers of geometric connected components
    of fibres of $f$ is locally constant.
    \end{lemma}
    
    \begin{proof}
    By Lemma \ref{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}
    the function $n_{X/S}$ is lower semincontinuous.
    For $s \in S$ consider the $\kappa(s)$-algebra
    $$
    A = H^0(X_s, \mathcal{O}_{X_s})
    $$
    By Varieties, Lemma
    \ref{varieties-lemma-proper-geometrically-reduced-global-sections}
    and the fact that $X_s$ is geometrically reduced
    $A$ is finite product of finite separable extensions of $\kappa(s)$.
    Hence $A \otimes_{\kappa(s)} \kappa(\overline{s})$ is a product
    of $\beta_0(s) = \dim_{\kappa(s)} H^0(E \otimes^\mathbf{L} \kappa(s))$
    copies of $\kappa(\overline{s})$. Thus
    $X_{\overline{s}}$ has $\beta_0(s) = \dim_{\kappa(s)} A$
    connected components. In other words, we have $n_{X/S} = \beta_0$
    as functions on $S$. Thus $n_{X/S}$ is upper semicontinuous by
    Derived Categories of Schemes, Lemma \ref{perfect-lemma-jump-loci-geometric}.
    This finishes the proof.
    \end{proof}

    Comments (1)

    Comment #2798 by BB on September 7, 2017 a 2:01 pm UTC

    The term "semicontinuous" is spelled incorrectly in the first sentence of the proof. Also, unless I'm mistaken, this term more frequently appears as hyphenated as "semi-continuous" elsewhere in the SP.

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