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IntegralClosure :: makeS2

makeS2 -- compute the S2ification of a reduced ring

Synopsis

Description

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 $R\subset S\subset {\rm frac}(R)$ 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 $P^3$

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..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..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

Caveat

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

See also

Ways to use makeS2 :

For the programmer

The object makeS2 is a method function with options.