ac.commutative algebra – On the use of the fundamental exact sequence of K”ahler differentials in a paper of Lyubeznik

Let $k$ be a field, $R := k(x_1, cdots , x_n)$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading, gives a “characteristic-free” proof of the finiteness of the length of the localized ring $R_f$, viewed as a $mathfrak D$-module over the ring $mathfrak D$ of $k$-linear differential operators of $R$:

I seem to be having the following problem with the proof of Proposition 3.1:

In the first paragraph of the proof (the one that basically shows the existence of a separating transcendence basis of a certain transcendental extension of a perfect field invoking the machinery of K”ahler differentials) the author seems to be considering the field of fractions $K$ of the domain $R/P$ where $P$ is a certain prime ideal of the polynomial ring $R$ and towards the end of the paragraph, he obtains (by relabeling) elements $x_{h+1},cdots , x_n$ such that the ring of K”ahler differentials $Omega_{K/k}$ is spanned as a $K$-vector space by ${d(x_{h+1}),cdots , d(x_n)}$, where $d$ is the natural map $R/P rightarrow Omega_{K/k}$. Thereafter, he considers the polynomial subring $mathscr R := k(x_{h+1},cdots , x_n)$ of $R$ and denoting by, $mathscr K$ its field of fractions, he speaks of the “fundamental exact sequence”
$$Omega_{mathscr K / k} otimes_{mathscr K} K longrightarrow Omega_{K/k} longrightarrow Omega_{K/mathscr K} longrightarrow 0.$$

Now, I have recently read very little about K”{a}hler differentials and from what I know, if $B$ and $C$ are algebras over a commutative ring $A$ such that there is an $A$-algebra homomorphism from $B$ to $C$ (giving $C$ the structure of a $B$-module), then we have an exact sequence of $C$-modules
$$Omega_{B/A} otimes_{B} C longrightarrow Omega_{C/A} longrightarrow Omega_{C/B} longrightarrow 0.$$
If this is indeed the result being used by Lyubeznik in the part quoted above, I seem to be having trouble figuring out what the $k$-algebra map from
$$mathscr K := text{Frac }k(x_{h+1},cdots , x_n) = k(x_{h+1},cdots , x_n) hspace{5mm} text{(which seems to play the role of $B$)}$$
to the ring
$$K:= text{Frac}(R/P) = text{Frac}left(k(x_1,cdots , x_n) big/ P right) hspace{5mm} text{(which seems to play the role of }Ctext{)}$$
should be.

I started with the $k$-algebra map
$$eta: mathscr R = k(x_{h+1},cdots , x_n) hookrightarrow R twoheadrightarrow R/P.$$
But now by the universal property of the total quotient ring, in order for this to factor through $mathscr K := text{Frac}(mathscr R)$, every nonzero element of $mathscr R$ must be a unit in $R/P$. This is clearly false if $Pmathscr R neq 0$, for any nonzero element of $Pmathscr R subset mathscr R$ maps to zero under $eta$. I apologize if I am missing something obvious but I would really appreciate some help in this regard. Thank you.

Addendum: It did occur to me that the perhaps the provided definition of $mathscr R$ is a typo (it is just defined in one place after all and there are other definitions building upon it) and perhaps what was really meant was that $$mathscr R:= k(x_{h+1},cdots , x_n)/Pk(x_{h+1},cdots , x_n),$$ which is still a domain. While it is intuitively clear that most of the other parts of the proof seem to go through with this choice of $mathscr R$, I have not yet been able to rigorously verify the same. Is this guess correct?