Page 17 of the following survey:
<http://arxiv.org/abs/1103.5380>
makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are automatically crepant.


Why is this true, or can someone point me to a reference?


Thanks.



Small resolutions are automatically crepant?
Let $X$ be a normal $\mathbb{Q}$-factorial variety, and let $f:Y\rightarrow X$ be a resolution. Then we can write 
$$K\_Y = f^{\*}K\_X+\sum a(E\_i,X)E\_i,$$
where the $E\_i$'s are $f$-exceptional divisor. If $f$ is small then the contracted locus is in codimension at least two. That is there is no $E\_i$, and $K\_Y = f^{\*}K\_X$.


Alternatively, one can consider the open immersion $i:X\_{reg}\rightarrow X$, and define $\omega\_{X}:=i\_\*\omega\_{X\_{reg}}$. Note that $\omega\_{X} = \mathcal{O}\_{X}(K\_{X})$. 


Furthermore, if $X$ is be a projective variety and $U=X∖Z$ is an open subset, $\mathcal{F}$ is a torsion-free coherent sheaf on $U$, $Z$ has codimension greater or equal that two, and $X$ is normal. Then $i\_{\*}\mathcal{F}$ is a coherent sheaf on $X$, where $i:U\rightarrow X$ is the inclusion. You can find this theorem here: J. P. Serre, *"Prolongement de faisceaux analytiques cohérents"*, Ann.Inst.Fourier 16, 363-374.


Therefore $\omega\_{X}:=i\_\*\omega\_{X\_{reg}}$ is a coherent sheaf on $X$. Now, let $f:Y\rightarrow X$ be a small resolution. Then $f^{\*}\omega\_{X}$ is a coherent sheaf on $Y$. Let $V\subset Y$ be the open subset where $f$ is an isomorphism. Then $\omega\_{Y|V}\cong (f^{\*}\omega\_{X})\_{|V}$. Now, $\omega\_{Y}$ and $f^{\*}\omega\_{X}$ are coherent sheaves on a smooth variety which are isomorphic outside of a closed subset of codimension at least two. Therefore $\omega\_{Y}\cong f^{\*}\omega\_{X}$.


