Let $f:X \to Y$ be a flat, projective morphism with $Y$ integral and every fiber of $f$ normal and integral. Let $F$ be a torsion-free, coherent sheaf on $X$ (not necessarily flat over $Y$). Then, is the function $y \mapsto \chi(F|\_{X\_y})$, upper semi-continuous, where $X\_y:=f^{-1}(y)$?


Also, given any discrete valuation ring $R$ and a morphism $g:\mathrm{Spec}(R) \to Y$ with the
generic point of $\mathrm{Spec}(R)$ mapping to the generic point of $Y$. Denote by $F\_R$ the pull-back of the $F$ to $X\_R$ via the morphism $g$. We know that if $F\_R$ is torsion-free, then $F\_R$ is flat over $\mathrm{Spec}(R)$, hence the Euler characteristic of $F\_K$ is the same as that of $F\_k$ (here $K$ and $k$ are the fraction field and residue field of $R$, respectively).


My question is: if $F\_R$ is *not* torsion-free, then is the support of the
torsion sub-sheaf of $F\_R$ contained in the closed fiber $X\_k$? If necessary, assume that the restriction of $F$ to the generic fiber $X\_\eta$ of $f$ is torsion-free.



Variation of Euler characteristic when the sheaf is not flat
Kollar in [his article](https://arxiv.org/pdf/1503.08694.pdf) mentions that in general Euler characteristic is lower semi-continuous without the assumption of flatness. However, there is no proof of this statement in the article.


