makeS2 -- compute the S2ification of a reduced ring

Synopsis

Usage:

(F,G) = makeS2 R
Inputs:
- R, a ring, an equidimensional reduced (or just unmixed and genericaly Gorenstein) affine ring
Optional inputs:
- Verbosity => an integer, default value 0, larger values give more information.
- Variable => ..., default value "w", Sets the name of the indexed variables introduced in computing the S2-ification.
Outputs:
- F, a ring map, $R \rightarrow S$, where $S$ is the so-called S2-ification of $R$
- G, a ring map, $frac S \rightarrow frac R$, giving the corresponding fractions

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 (------------------------------------------------------------------
                                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 R,
                                            
      - a*d)                                
                                            
     ------------------------------------------------------------------------
         /                              ZZ                                   
         |                             ---[w   , a..d]                       
         |                             101  0,0                              
     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         0,0 
     ------------------------------------------------------------------------
           \
           |
           |   b*d
     ------|, {---, a, b, c, d}))
           |    c
     - a*d)|
           /

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

Ways to use makeS2 :

makeS2(Ring)

For the programmer