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hbghlyj

Kähler differentials in complete function fields

let K:=k((t)) a field of formal power-series and let be (K|k) denote the differential-modul of K|k and d:K->(K|k) the universal derivation, then we should have the nice identity

df = f' dt ,

where f is from K and f' is formal derivation of f after t. this question may be too trivial, but can the identity be derived only by the universal property of the pair
(d, (K|k)) or do we need a continuity argument here?
if K were given as K=k(t) we trivialy would get our relation, but what is in the case of formal power series?

hbghlyj

Question about thm 1.1.23

Question about thm 1.1.23
etengan
Member
Posted on 15 March 2003 at 12:54.35 PM

Hi,

I am a new user to this forum and I've got a question about thm 1.1.23 on page 12. Can anyone help me?

On equation (page 13)

(**) dim_F O[y]/PO[y] \le dim_F R/PR,

how do I know this is true? I suspect that O[y]/PO[y] injects into R/RP, although I don't know how to prove this.

I appreciate any comments on this issue.

Bye,
ET

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Re: Question about thm 1.1.23
david
Moderator
Posted on 25 March 2003 at 10:06.43 AM

This is a good question, and it looks like the exposition is incomplete at this point. I haven't had time to think about it very much, but perhaps the following argument fills the gap.

The missing fact that we need in order to conclude that O[y]/PO[y] injects via the natural map to R/PR is that PR \cap O[y] = PO[y]. Since one inclusion is obvious, we need only show that PR \cap O[y] \subseteq PO[y]. Because O[y]/PO[y] is a direct sum of fields by (*), PO[y] is the intersection of maximal ideals P_1,...,P_r' of O[y]. Since R is integral over O[y], there are prime ideals Q_1,...,Q_r 'of R with Q_i lying over P_i for each i. (This is essentially a consequence of Nakayama's Lemma -- see, e.g., prop. 4.15 of Eisenbud). Put I := \cap_i Q_i, then PR\subseteq I, and I \cap O[y] = \cap_iP_i = PO[y] as required.