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I realise this is a strange question but my supervisor hasn't been replying to my emails. I am trying to understand the relationship between the transcendental dimension of irreducible affine varieties (as in Atiyah and MacDonald, Chapter 11) and the notion of Kahler differentials. Are there any chapters in Matsumura, Commutative Ring Theory, which explain the connection?

Any help would be appreciated! If I should be looking elsewhere, please let me know.

coolpenguin
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    Perhaps this is more immediate to those with more expertise than me, but what exactly is the connection that you want explained? I don't see any obvious relationship between the two ideas other than a few modest theorems. (For instance, as you can check on Wikipedia, if $X$ is a smooth variety over a field $k$, then the sheaf of differentials $\Omega_{X/k}$ is a locally free $\mathcal{O}_{X}$-module of rank $\dim(X)$. Among other things, this gives rise to the notion of the canonical divisor.) It would help if you could clarify more precisely what you want elucidated. – Alex Wertheim Sep 01 '21 at 23:52
  • I have started another question here, which is more specific here https://math.stackexchange.com/questions/4249517/module-of-kahler-differentials-of-an-affine-variety. Please comment – coolpenguin Sep 13 '21 at 18:02

2 Answers2

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For a variety over a field, the sheaf of Kahler differentials is coherent and thus has a rank, say $n$. In general, one has $n\geq \dim X$ and equality in characteristic zero.

Mohan
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Question: "Any help would be appreciated! If I should be looking elsewhere, please let me know."

Answer: In Hartshorne Theorem II.8.15 you find the following result: If $X:=Spec(A)$ where $A$ is a finitely generated algebra over an algebraically closed field $k$ where $krdim(X)=n$, then $\Omega^1_{A/k}$ is a locally trivial $A$-module of rank $n$ iff $X$ is nonsingular (which is iff $A$ is regular). You also find some results and references to results in Matsumura's book. Here $krdim(X)$ (or $krdim(A)$) is the "Krull dimension" of $X$ (or of $A$) which is defined in terms of prime ideals.

hm2020
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