# 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 $$K:= \text{Frac}(R/P) = \text{Frac}\left(k[x_1,\cdots , x_n] \big/ P \right) \hspace{5mm} \text{(which seems to play the role of }C\text{)}$$ 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 $$P\mathscr R \neq 0$$, for any nonzero element of $$P\mathscr R \subset \mathscr R$$ 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?

• Why don't you ask him? May 3, 2021 at 22:15
• Thank you Sir, I will try doing that then. I have never had the opportunity to email authors of papers just for clarification before, hence I didn't consider it at first. May 3, 2021 at 22:33