A ring S satisfies Serre’s S2 condition if every codimension 1 ideal contains a nonzerodivisor and every principal ideal generated by a nonzerodivisor is equidimensional of codimension one. If R is an affine reduced ring, then there is a unique smallest extension R⊂S⊂frac(R) satisfying S2, and S is finite as an R-module.
Uses the method of Vasconcelos, "Computational Methods..." p. 161, taking the idealizer of a canonical ideal.
There are other methods to compute S, not currently implemented in this package. See for example the function (S2,Module) in the package "CompleteIntersectionResolutions".
We compute the S2-ification of the rational quartic curve in P3
i1 : A = ZZ/101[a..d]; |
i2 : I = monomialCurveIdeal(A,{1,3,4}) 3 2 2 2 3 2 o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o2 : Ideal of A |
i3 : R = A/I; |
i4 : (F,G) = makeS2 R ZZ ---[w , a, b, c, d] 101 0,0 o4 = (map(-------------------------------------------------------------------------,R,{a, b, c, d}), map(frac 2 2 2 (b*c - a*d, w d - c , w c - b*d, w b - a*c, w a - b , w - a*d) 0,0 0,0 0,0 0,0 0,0 ---------------------------------------------------------------------------------------------------------------------------- / ZZ \ | ---[w , a, b, c, d] | | 101 0,0 | b*d R,frac|-------------------------------------------------------------------------|,{---, a, b, c, d})) | 2 2 2 | c |(b*c - a*d, w d - c , w c - b*d, w b - a*c, w a - b , w - a*d)| \ 0,0 0,0 0,0 0,0 0,0 / o4 : Sequence |
Assumes that first element of canonicalIdeal R is a nonzerodivisor; else returns error. The return value of this function is likely to change in the future