I am reading O. Fujino's book [*Iitaka Conjecture*](https://link.springer.com/book/10.1007/978-981-15-3347-1). 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.



Inclusion of (pulling back of) dualizing sheaves under normalization
$X$ 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.


