Lemma 49.11.3. With notation as in Example 49.11.2 there is an open subscheme U_ d \subset X_ d with the following property: a morphism of schemes X \to X_ d factors through U_ d if and only if Y_ d \times _{X_ d} X \to X is syntomic.
Proof. Recall that being syntomic is the same thing as being flat and a local complete intersection morphism, see More on Morphisms, Lemma 37.62.8. The set W_ d \subset Y_ d of points where \pi _ d is Koszul is open in Y_ d and its formation commutes with arbitrary base change, see More on Morphisms, Lemma 37.62.21. Since \pi _ d is finite and hence closed, we see that Z = \pi _ d(Y_ d \setminus W_ d) is closed. Since clearly U_ d = X_ d \setminus Z and since its formation commutes with base change we find that the lemma is true. \square
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