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On a scheme, being normal means that each stalk of the structure sheaf is a integrally closed domain. Being regular means that each stalk of the structure sheaf is a regular local ring.

As for a local ring, being regular or being integrally closed does not imply another.

What is their connection with each other and classical/usual intuition of being smooth(being regular on stalk of each closed points)?

Moreover, is there a smooth/regular variety which is not normal?

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I am especially interested in the meaning of being normal in dimension no less than 2.(on curves being normal is pretty clear now.) –  7-adic Apr 1 '10 at 8:41
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See this question: mathoverflow.net/questions/12688/nonsingular-normal-schemes –  Hailong Dao Apr 1 '10 at 13:32
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2 Answers 2

Dear 7-adic, yes there is an implication between the two notions.

For a local ring, regular implies normal. Actually Auslander and Buchsbaum proved in 1959 that a regular local ring is a UFD and it is an easy result that a UFD (local or not) is integrally closed. Serre then gave a completely different proof. He proved that regular is equivalent to having finite global (=homological) dimension . This finiteness means that any module over the ring has a finite projective resolution. I have heard it claimed that this was the beginning of the acknowledgment of the importance of homological algebra in commutative algebra.

An example.The cone $z^2=xy$ in affine 3-space (over a field, say) is normal but not regular: its very equation suggests that we don't have the UFD property and this intuition can be converted into a rigorous proof. Normality is a weak form of regularity. The two concepts coincide in dimension one but not in higher dimensions: the quadratic cone above shows this in dimension two.

Finally, smoothness is even stronger: it is a relative concept meaning regular and remaining regular after base change.

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You seem a bit confused. A regular* local ring is a UFD hence integrally closed. In other words, regular implies normal. See for instance

http://www.math.iitb.ac.in/atm/caag1/jayanthan.pdf

for a relatively elementary algebraic treatment.

*: I had previously included Noetherian here, but after checking on this I see I was being overly careful: it is part of the definition of a regular local ring that it be Noetherian.

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Eisenbud's book "Commutative Algebra (with a view to Algebraic Geometry)" has a comprehensible and well-presented treatment too, building it up from scratch. –  Ravi Vakil Apr 4 '10 at 21:00
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