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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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Call the morphism $\pi$. I think you're good if $R\pi_*F$ is quasi-isomorphic to a perfect complex. This is true if $F$ is flat over $Y$, and as far as I can tell is the only non-trivial input in this case. – Keerthi Madapusi Pera Feb 21 '13 at 0:13
@Keerthi: If $Y$ is regular then any (bounded) complex with coherent cohomology is (quasi-isomorphic to) a perfect complex. – ulrich Feb 21 '13 at 3:59
Keerti: I don't think your claim is true even for derived tensor products; $F$ could be a line bundle in which case the derived tensor product is just the usual restriction to the fibre. For an explicit example, let $Y$ be a smooth surface, $X$ the blowup of a point and $F = \mathcal{O}_X(E)$, where $E$ is the exceptional divisor. For all $y$ not equal to the blown up point, $H^0$ is one dimensional, but it is zero dimensional for the blown up point. – ulrich Feb 22 '13 at 5:09
In EGA III, Theoreme 7.7.5, a semi-continuity theorem is proved for complexes of sheaves that are flat over $Y$. – ulrich Feb 22 '13 at 5:30
ulrich--That reference seems to answer the question as well as one could hope to. Maybe you should put it down as an answer? – Keerthi Madapusi Pera Feb 22 '13 at 12:34

Since no one really bothered to write down an answer, I think it might be worth publicising the paper by Heinrich Hartmann "Cusps of Kähler moduli space and stability conditions on K3 surfaces." Proposition 6.4 might be what you want (and before that he's got some useful non tor-independent base change theorems).

A link to the paper would have been useful.

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