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.


1 Answer 1


Kollar in his article 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.


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