The Stacks project

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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