The Stacks project

Lemma 37.27.3. Let $f : X \to Y$ be a morphism of schemes. Let

\[ n_{X/Y} : Y \to \{ 0, 1, 2, 3, \ldots , \infty \} \]

be the function which associates to $y \in Y$ the number of irreducible components of $(X_ y)_ K$ where $K$ is a separably closed extension of $\kappa (y)$. This is well defined and if $g : Y' \to Y$ is a morphism then

\[ n_{X'/Y'} = n_{X/Y} \circ g \]

where $X' \to Y'$ is the base change of $f$.

Proof. Suppose that $y' \in Y'$ has image $y \in Y$. Suppose $K \supset \kappa (y)$ and $K' \supset \kappa (y')$ are separably closed extensions. Then we may choose a commutative diagram

\[ \xymatrix{ K \ar[r] & K'' & K' \ar[l] \\ \kappa (y) \ar[u] \ar[rr] & & \kappa (y') \ar[u] } \]

of fields. The result follows as the morphisms of schemes

\[ \xymatrix{ (X'_{y'})_{K'} & (X'_{y'})_{K''} = (X_ y)_{K''} \ar[l] \ar[r] & (X_ y)_ K } \]

induce bijections between irreducible components, see Varieties, Lemma 33.8.7. $\square$

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