76.30 Connected components of fibres
This section is the analogue of More on Morphisms, Section 37.28.
Lemma 76.30.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let
n_{X/Y} : |Y| \to \{ 0, 1, 2, 3, \ldots , \infty \}
be the function which associates to y \in Y the number of connected components of X_ k where \mathop{\mathrm{Spec}}(k) \to Y is in the equivalence class of y with k algebraically closed. 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. Let \mathop{\mathrm{Spec}}(k') \to Y' be in the equivalence class of y' with k' algebraically closed. Then we can choose a commutative diagram
\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[rd] & \mathop{\mathrm{Spec}}(k') \ar[r] & Y' \ar[d] \\ & \mathop{\mathrm{Spec}}(k) \ar[r] & Y }
where K is an algebraically closed field. The result follows as the morphisms of schemes
\xymatrix{ X'_{k'} & (X'_{k'})_ K = (X_ k)_ K \ar[l] \ar[r] & X_ k }
induce bijections between connected components, see Spaces over Fields, Lemma 72.12.4. To use this to prove the function is well defined take Y' = Y.
\square
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