# Variation of Euler characteristic when the sheaf is not flat

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.