## The Jacobian ideal generates the socle of a complete intersection

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This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here: http://tinyurl.com/2967eov

I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ is a complete intersection and $dim_k A$ is not divisible by $char(k)$ then the Jacobian ideal generates the socle of $A$".

I am looking for a proof of this theorem. Vasconcelos references three places to look for one. One is a result of Tate - I have looked at this. One is supposed to be in Kunz's - Introduction to commutative algebra and algebraic geometry" - I could not find a result similar to this in there (it's not a pointed reference). Finally there is a Scheja-Storch paper linked below. http://www.reference-global.com/doi/abs/10.1515/crll.1975.278-279.174 I am specifically looking for a proof similar to Scheja-Storch (Tate seems to use a different approach), but the above paper is in German and I am not fluent at it. It's probably unlikely, but if anyone has an english reference on this proof, I would really appreciate it.

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@Graham: Thanks, fixed. – Timothy Wagner Nov 18 2010 at 2:52
This doesn't probably help really (since it's not a proof), but this seems to be an exercise on page 382 of Kunz's book, "Kahler differentials". Perhaps reading near there will suggest the proof? – Karl Schwede Nov 18 2010 at 4:16
Ah, thanks Karl. I guess Vaconcelos referenced a different Kunz's book in error. But I checked two versions and in both the above one was cited. I shall look into the Kahler differentials one. Thank you. – Timothy Wagner Nov 18 2010 at 4:42

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This paper by Eisenbud, Huneke, and Vasconcelos (msri.org/people/staff/de/papers/pdfs/1992-001.pdf) attributes the result to Scheja and Storch, Cor. 4.7 of the paper you linked, but gives no indications of proof. Aha! Prop.2 of this paper by Eisenbud (projecteuclid.org/euclid.bams/1183541138) attributes it 'essentially' to Berger, and has a sketch of a proof. – Graham Leuschke Nov 18 2010 at 14:11
@Graham: Thanks a lot for the references. I will have to look more closely at the second one. – Timothy Wagner Nov 19 2010 at 23:39

## 1 Answer

I'm promoting this comment to an answer, since it appears no one else is jumping in with a proof. Prop. 2 of this paper by Eisenbud attributes it 'essentially' to Berger, and has a sketch of a proof.

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