Let $X$ be a noetherian, projective scheme, $\mathcal{F}$ be a reflexive sheaf on $X$ pure of dimension $\dim(X)$ and $Y \subset X$ be a closed subscheme of $X$. Is it possible that the pull-back of $\mathcal{F}$ to $Y$ is again reflexive?

  • 2
    $\begingroup$ Of course it is possible. For instance, if $\mathcal{F}$ is a vector bundle (=locally free sheaf), so is its pullback (=restriction to $Y$). $\endgroup$ Apr 28, 2015 at 11:49
  • $\begingroup$ @Polizzi: Are there cases when $\mathcal{F}$ is not a locally free sheaf? $\endgroup$
    – user43198
    Apr 28, 2015 at 11:55
  • $\begingroup$ Yes. A reflexive sheaf on $X$ is locally free outside a closed subset $D$ of codimension $\geq 3$. Now, for instance, take $Y$ disjoint from $D$. $\endgroup$ Apr 28, 2015 at 12:09
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    $\begingroup$ @Polizzi: This is true only if $X$ is regular. $\endgroup$
    – user43198
    Apr 28, 2015 at 12:10
  • $\begingroup$ True, I'm considering the simplest case. $\endgroup$ Apr 28, 2015 at 12:12

1 Answer 1


Let $X$ be a noetherian integral scheme and $\mathscr F$ a coherent sheaf on $X$. If $\mathscr F$ is reflexive, then $\mathscr F\big|_H$ is torsion-free for any Cartier divisor $H \subset X$, and $\mathscr F\big|_H$ is reflexive if $H$ is a general element of a basepoint-free linear system (the notion of generality here of course depends on $\mathscr F$). If $\mathscr F$ is just torsion-free, then $\mathscr F\big|_H$ is still torsion-free for $H$ general.

Some references for this are Huybrechts-Lehn, ''The Geometry of Moduli Spaces of Sheaves'', Cor. 1.1.14, and Greb-Kebekus-Peternell, ''Étale fundamental groups of klt spaces'', Prop. 5.1 and 5.2.


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