The existence of the natural morphism $f_*\omega_{\widetilde X}\rightarrow\omega_X$.

Question

Let $X$ be a normal variety, and denote its canonical sheaf by $\omega_X$. Consider the resolution of singularities $f:\widetilde X\rightarrow X$, I have been told that there always exists a natural morphism $f_*\omega_{\widetilde X}\rightarrow\omega_X$. My question is how can we describe this morphism concretely? Any reference and suggestion would be appreciated. Thanks a lot!

Answer 1

Given $U\subset X$, I'll tell you how to construct the map of sections over $U$. Let $W\subset U$ be smooth with complement at least codimension two. Then $f^{-1}(W)\to W$ is an isomorphism, and $\omega_X(W)=\omega_X(U)$, since the canonical sheaf on a normal scheme is determined by its behavior away from a set of codimension two. So the map $$f_*(\omega_{\widetilde{X}})(U)\stackrel{f_*res_{f^{-1}(U),f^{-1}(W)}}{\to} f_*(\omega_{\widetilde{X}})(W)\stackrel{\sim}{\to} \omega_{X}(W) \stackrel{=}{\to} \omega_X(U)$$ does the job.