# R2 and S3 for rings.

For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition that every prime P of codimension at least 2 satisfies depth R_P \geq 2. Likewise, there's a similar condition for whether or not R is reduced: R is reduced iff R satisfies R0 and S1. Following the pattern, it seems like Rn and S(n+1) should be equivalent to some desirable property for rings.

Now, there's a nice and canonical way to take any ring and create a reduced (or normal) ring out of it, and this is something we should not expect to extend to higher dimensions, since we know that resolving singularities is (a) hard and (b) not very canonical. That aside, surely we ought to be able to say something nontrivial about the next case, ie, R2 and S3?

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The operations of reduction (making a ring R0 and S1) and normalization (R1 and S2) stay in the category of affine schemes. Supernormalization can not: The non-R2 singularity k[x,y,z]/xz-y^2 can not be removed by any birational proper map from an affine domain. If you remove "affine", you have probably returned to the problem of resolution of singularities. I'm not sure what happens if you remove birational.

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The word you're looking for is "supernormal". A couple sample results: If R is supernormal, then the map on divisor class groups Cl(R) \to Cl(R[[t]]) is an isomorphism (Danilov, Griffith). If R is a supernormal UFD, then the completion \hat{R} is a UFD as well (Flenner).

I don't know of any work on "supernormalization" -- a quick search in MathSciNet doesn't turn up anything.

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If you weaken birational to just proper and generically finite ("alterations"), then you can do this thanks to de Jong's theorems. In fact, he allows you to construct generically etale alterations. Also, if you just try to make things Cohen-Macaulay (S_k for all k) while forgetting about regularity, then there is a theorem of Kawasaki that says essentially everything admits a proper birational map from something Cohen-Macaulay. It seems hard to do better: a 3-dimensional normal non-CM singularity in characteristic 0 does not admit a finite map from something CM (the local cohomology of singularity is a summand of that of the cover for any normal ring in char 0).

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