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I was looking for properties of reflexive sheaves on a variety. Suppose $X\rightarrow Y$ is etale outside codimension two subset. Say $X$ is smooth. Is the pullback of a reflexive sheaf on $Y$, reflexive on $X$.

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    $\begingroup$ Dear John, Outside of codim'n two on $X$ or on $Y$? Regards, $\endgroup$
    – Emerton
    Apr 16 '13 at 2:29
  • $\begingroup$ Karl's answer modifies to an counterexample even if $X$ and $Y$ are isomorphic in codim $2$, for example, consider the blowup $k[x,y,z,w]/(xy-zw)$ along the plane $x=z=0$, then locally the map reads $k[x,y,z,w]/(xy-zw)\to k[x,z,w], x\mapsto x, y\mapsto zw, z\mapsto zx, w\mapsto w$, the pullback of the ideal $(y,z)$ has an nontrivial element $y\otimes x-z\otimes w$ annihilated by $yz$. $\endgroup$
    – Qixiao
    Oct 17 '19 at 2:21
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I think this is not true. For example, what if $X \to Y$ is a resolution of singularities, that's even an isomorphism outside a set of codimension 2. The pullback of reflexive sheaves is definitely not reflexive.

For example, consider the blowup of the origin in $R = k[x,y,z]/(xy-z^2) = k[a^2, b^2, ab]$. One of the charts in that blowup is $k[a^2, b/a]$. The pullback of the reflexive sheaf $(a^2, ab)$ in that chart is the tensor product $(a^2, ab) \otimes_R k[a^2, b/a]$. Then the element $$a^2 \otimes (b/a) - ab \otimes 1$$ is clearly torsion and non-zero.

What people do frequently do is the reflexive pullback. In other words, pullback and then reflexify.

From the point of view of singularities, you might look at a paper of de Fernex and Hacon on singularities in non-Q-Gorenstein rings. There are other sources I can point you towards (say Hassett and Kov\'acs) if you are more interested in moduli types of applications.

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