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This is an extremely basic (and surely amateurish) question that might be about derived geometry.

In usual algebraic geometry, if we have a flat projective morphism $f:X \to S$ with $S$ integral, and fibers $X_s = f^{-1}(s)$, then certain invariants are constant in the family $\{X_s:s \in S\}$ (in some sense, the Hilbert polynomial is the only one, and the others can be obtained from it).

Now let's drop the flatness assumption. Obviously, nothing is constant any more in a non-flat family.

Question. Is there anything constant for non-flat families if we take the derived fibers instead? Perhaps under some assumptions, or phrased differently?

I am almost certain that the answer is either negative or extremely well-known.

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    $\begingroup$ Replacing dim $H^0$ with the Euler characteristic should work whenever it's finite. $\endgroup$
    – Jon Pridham
    Commented Jan 24 at 9:11

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