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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⊂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(-------------------------------------------------------------------
                               2                                     2   2   
          (b*c - a*d, w   d - c , w   c - b*d, w   b - a*c, w   a - b , w    
                       0,0         0,0          0,0          0,0         0,0 
     ------------------------------------------------------------------------
                                     
                                     
                                     
     ------,R,{a, b, c, d}), map(frac
                                     
     - a*d)                          
                                     
     ------------------------------------------------------------------------
           /                           ZZ                                    
           |                          ---[w   , a, b, c, d]                  
           |                          101  0,0                               
     R,frac|-----------------------------------------------------------------
           |                     2                                     2   2 
           |(b*c - a*d, w   d - c , w   c - b*d, w   b - a*c, w   a - b , w  
           \             0,0         0,0          0,0          0,0         0,
     ------------------------------------------------------------------------
             \
             |
             |  b*d
     --------|,{---, a, b, c, d}))
             |   c
       - a*d)|
     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 :