Let me state the question for rings (rather than schemes) for simplicity. Let $R$ be a commutative ring with unit and $A$ an $R$-algebra of finite presentation. Recall that $R\to A$ is called a syntomic morphism if it is flat and for each morphism $R\to k$ to a field, the ring $A\otimes_R k$ is a local complete intersection in the sense of commutative algebra.
Do you know a simple functorial criterion for $R\to A$ to be syntomic, in the spirit of the functorial criterion for smoothness saying that $R$-morphisms $A\to B/I$ lift to $A\to B$ whenever $I\subset B$ is a square-zero ideal?
Variants of such criteria in this functorial flavour, or examples in the literature, are welcome. Variants where $R\to A$ is already known to be flat are also welcome (for smooth morphisms, flatness somehow comes for free but it is sort of a small miracle).
My motivation to ask this question is that I have a morphism which I can prove is syntomic, but only indirectly, using smoothness in an indirect way. I'm wondering whether a direct attack would be possible.
