# 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 $$R$$: https://www.sciencedirect.com/science/article/pii/S002240491000263X

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?