In algebraic geometry, the very useful semicontinuity theorem tells you the following:

Let $X \to Y$ be a projective morphism of schemes, and $F$ a coherent sheaf on $X$ which is flat over $Y$. Then the dimensions of the cohomology on fibers, $h^i(X_y)$ are upper semicontinuous functions of $y$, i.e., the locus where in $Y$ where $h^i(X_y) \ge n$ is Zariski closed for any $n$.

What condition on a complex $F$ with coherent cohomology ensures that $\dim \mathbb{H}^i(X_y, F \otimes^L \mathcal{O}_{X_y})$ is an upper semicontinuous function of $y$?

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