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 $X$.

1$\begingroup$ Dear John, Outside of codim'n two on $X$ or on $Y$? Regards, $\endgroup$– EmertonApr 16, 2013 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 $k[x,y,z,w]/(xyzw)$ along the plane $x=z=0$, then locally the map reads $k[x,y,z,w]/(xyzw)\to k[x,z,w], x\mapsto x, y\mapsto zw, z\mapsto zx, w\mapsto w$, the pullback of the ideal $(y,z)$ has an nontrivial element $y\otimes xz\otimes w$ annihilated by $yz$. $\endgroup$– QixiaoOct 17, 2019 at 2:21
1 Answer
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 $R = k[x,y,z]/(xyz^2) = k[a^2, b^2, ab]$. 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 nonzero.
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 nonQGorenstein 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.