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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.

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    $\begingroup$ Note that your resolution is an isomorphism over the smooth locus. When you pullback the dualizing sheaf, you get a coherent sheaf that agrees with the dualizing sheaf of the resolution outside the exceptional set. Since your dualizing sheaf on a regular variety is invertible, it is $S_2$. Now, for example, you can use Corollary 1.18 in the note: personal.psu.edu/kes32/Notes/GeneralizedDivisors.pdf $\endgroup$
    – Honglu
    Commented May 9, 2014 at 4:24
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    $\begingroup$ I think that originally the notion of crepant resolution $f\colon Y\to X$ was given when K_X is Q-Cartier, so that it makes sense to consider discrepancies of exceptional divisors in a resolution. If K_X is not Q-Cartier, it may very well happen that f is small and contracts curves having non-zero intersection with K_Y. $\endgroup$
    – Cinzia Casagrande
    Commented May 9, 2014 at 15:16

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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 $\omega_{Y}\cong f^{*}\omega_{X}$.

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