Inclusion of (pulling back of) dualizing sheaves under normalization

Question

I am reading O. Fujino's book Iitaka Conjecture. In page 42, Lemma 3.1.19 he restated one result due to Viehweg to use the base change arguments.

There exists some details in the proof of Step 2 in Lemma 3.1.19 that I can't figure out as follows:

Let $X$ be a reduced Gorenstein analytic space which may be reducible, denote by $$\nu:\widetilde X\rightarrow X$$ the normalization of $X$. Then he claim that one can get an inclusion $$\iota:\omega_{\widetilde X}\hookrightarrow{\nu^*\omega_X}.$$

$\color{red}{\bullet}$ Thanks to relative trace map (one reference is Kollar-Mori's book Prop. 5.77), we have a natural map $$\textrm{Tr}:\nu_*\omega_{\widetilde X}\rightarrow \omega_X.$$ $\color{blue}{\bullet}$ As $\nu$ is finite, we have the following exact sequence: $$0\rightarrow\mathcal T\rightarrow \nu^*\nu_*\omega_{\widetilde X}\rightarrow \omega_{\widetilde X}\rightarrow 0,$$ here the sheaf $\mathcal T$ denotes the torsion part of $\nu^*\nu_*\omega_{\widetilde X}.$ These two steps seem suitable for me. However, he then said that $\iota$ holds thanks to $\color{red}{\bullet}$ and $\color{blue}{\bullet}$. This is somewhat too quick for me.

Now I will state some of my thoughts on how to get $\iota$.

Pulling back $\mathrm{Tr}$ by $\nu$, we get the morphism $$\mu: \nu^*\nu_*\omega_{\widetilde X}\rightarrow \nu^*\omega_X.$$

Here appears my first question,

Q1: Do $\mu$ must factor through $\omega_\widetilde X$? If so, why?

If Q1 holds, we indeed have $$\nu^*\nu_*\omega_{\widetilde X}\xrightarrow{\alpha} \omega_{\widetilde X} \xrightarrow{\iota} \nu^*\omega_X$$ with $\mu=\iota\circ\alpha.$

Here appears my second question,

Q2: do we have $\mathrm{Ker} {\mu} \subseteq \mathrm{Ker} \alpha=\mathcal T$? If so, why?

The last question is that

Q3: how is the sheaf map $\mu$ defined explicltly?

Thanks in advance.

Answer 1

$X$ is Gorenstein, so $\omega_X$ is a line bundle and hence so is $\nu^*\omega_X$. In particular, it is torsion-free and hence $\mu(\mathcal T)=0$, so $\mu$ indeed factors through $\omega_{\widetilde X}$. As $X$ is reduced, $\nu$ is generically an isomorphism, so ${\rm Ker}\ \iota$ is a torsion sheaf, and then because, as Leo pointed out, $\omega_{\widetilde X}$ is a torsion-free sheaf, it has to be zero. So, $\iota$ is injective as claimed.

In case your second question is still a question: both morphisms map to a torsion-free sheaf and as $\nu$ is generically an isomorphism, so are $\mu$ and $\alpha$. This means that the kernel of each morphism is exactly the torsion subsheaf of $\nu^*\nu_*\omega_{\widetilde X}$.

I am not really sure what you are asking in Q3. $(\ )^*$ is a functor, so it is defined on morphisms. If you look up its definition for sheaves anywhere, it should be clear how it is defined on morphisms.