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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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