Let $X$ be an irreducible scheme over some base field $k$. Consider the sheaf of Kähler differentials $\Omega_{X/k}$. Let $w: \Omega_{X/k} \to j_* \Omega_{K(X)/k}$ be natural map induced by enbedding of the generic point $j: \eta_X \to X$. As $X=\overline{ \{\eta_X\}}$ is contained in every open subscheme of $X$, $j_* \Omega_{K(X)/k}$ is just the constant sheaf of $K(X)$-module $\Omega_{K(X)/k}= \Omega_{X, \eta_X}$.
Question: Are there rather weak requirements known when $w: \Omega_{X/k} \to j_* \Omega_{K(X)/k}$ is injective, ie that $\Omega_{X/k}$ that is torstion free.
Of course, that it's hopeless to ask such kind of question for every coherent sheaf $\mathcal{F}$, but on other hand $\Omega_{X/k} $ is a pretty canonical object which carries a lot intrinsic information about geometry of $X$. For example locally freeness of it is equivalent to smoothness of $X$.
What kind of information about $X$ would encode torsion freeness of $\Omega_{X/k} $ ?
Motivation: It is known that when $X$ is smooth, $H^0(X, \Omega_{X/k})$ are birational invariants. A rather elementar argument (...there are a lot of other ways to see it) is for any birational map $f: Y \to X$ to consider the composition $X \dashrightarrow Y \xrightarrow{\text{f}} X $ with left map the rational pseudoinverse to $f$.
We get following canonical diagram
$$ \require{AMScd} \begin{CD} \Omega_{X/k}(X)=H^0(X, \Omega_{Y/k}) @>f^*>> H^0(Y, \Omega_{Y/k}) @>(f^{-1})^*>> H^0(X, \Omega_{X/k}) \\ @VwVV @VwVV @VwVV\\ j_* \Omega_{K(X)/k}(X)=\Omega_{K(X)/k} @>=>> \Omega_{K(Y)/k} @>=>> \Omega_{K(X)/k} \end{CD} $$
Lower row are just identities by assumed birationality, and if $w$ are injective, we would get that above rows are isos too by dimesnion count (... ok, let assume that that the $k$-vec spaces above are finitely generated, eg if $X,Y$ proper).
The advantage of this approach is that we would not need to assume $X,Y$ to be smooth. It would be weaker.
Indeed, if $X$ is smooth, then $\Omega_{X/k}$ locally free, and so $w$ is just tensoring of $O_X \to j_* O_{X, \eta_X}$ with $\Omega_{X/k}$. The latter is injective as smooth $X$, especially reduced, and so integral =irreducible + reduced.
So the question is are there are weaker conditions when $w$ is injective then the "overkill condition" when $X$ is smooth (which gives nothing new)?
This would as consequence imply that for those $X$ $H^0(X, \Omega_{X/k})$ is birational invriant even if $X$ not smooth.
