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Actually much more is true. In fact there is the following result(see , whose proof can be found in

[Grothendieck - Dieudonné: EGA I1 (Elements de Géométrie Algebrique), Proposition 8.4.5 p. 351)page 351].

Proposition. Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any torsion-free $\mathcal{O}_X$-module $\mathcal{F}$, the push-forward $f_* \mathcal{F}$ is a torsion-free $\mathcal{O}_Y$-module.

Kollar's result is much deeper since he proves the torsion-freeness of the higher direct images $R^if_* \omega_X$.

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Actually much more is true. In fact there is the following result (see EGA I, Proposition 8.4.5 p. 351).

Proposition. Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any torsion free torsion-free $\mathcal{O}_X$-module $\mathcal{F}$, the push-forward $f_* \mathcal{F}$ is a torsion free torsion-free $\mathcal{O}_Y$-module.

The very strength of

Kollar's result lies in the fact that is much deeper since he proves the torsion-freeness of the higher direct images $R^if_* \omega_X$.

1

Actually much more is true. In fact there is the following result (see EGA I, Proposition 8.4.5 p. 351).

Proposition. Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any torsion free $\mathcal{O}_X$-module $\mathcal{F}$, the push-forward $f_* \mathcal{F}$ is a torsion free $\mathcal{O}_Y$-module.

The very strength of Kollar's result lies in the fact that he proves the torsion-freeness of the higher direct images $R^if_* \omega_X$.