Lemma 70.6.3. With notation and assumptions as in Situation 70.6.1. If
f is smooth,
f_0 is locally of finite presentation,
then f_ i is smooth for some i \geq 0.
Proof. Choose an affine scheme V_0 and a surjective étale morphism V_0 \to Y_0. Choose an affine scheme U_0 and a surjective étale morphism U_0 \to V_0 \times _{Y_0} X_0. Diagram
\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }
The vertical arrows are surjective and étale by construction. We can base change this diagram to B_ i or B to get
\vcenter { \xymatrix{ U_ i \ar[d] \ar[r] & V_ i \ar[d] \\ X_ i \ar[r] & Y_ i } } \quad \text{and}\quad \vcenter { \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }
Note that U_ i, V_ i, U, V are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms U_ i \to V_ i is U \to V. Recall that X_ i \to Y_ i is smooth if and only if U_ i \to V_ i is smooth and similarly X \to Y is smooth if and only if U \to V is smooth (Morphisms of Spaces, Definition 67.37.1). Since f_0 is locally of finite presentation, so is the morphism U_0 \to V_0. Hence the lemma follows from Limits, Lemma 32.8.9. \square
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