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Seminormalization :: seminormalize

seminormalize -- seminormalize a reduced ring

Synopsis

Description

This seminormalizes a reduced ring and outputs the map from the original ring to the seminormalization .

i1 : R = QQ[x,y]/ideal(x^3 - y^2);
i2 : L = seminormalize(R)

                 QQ[Yy , Yy , Yy ]            
                      0    1    2             
o2 = {---------------------------------------,
         2                        2           
      (Yy  - Yy , Yy Yy  - Yy , Yy  - Yy Yy ) 
         2     1    1  2     0    1     0  2  
     ------------------------------------------------------------------------
                    QQ[Yy , Yy , Yy ]                          
                         0    1    2                           
     map(---------------------------------------,R,{Yy , Yy }),
            2                        2                1    0   
         (Yy  - Yy , Yy Yy  - Yy , Yy  - Yy Yy )               
            2     1    1  2     0    1     0  2                
     ------------------------------------------------------------------------
                                                                             
                               QQ[Yy00000RE1, xRE1, yRE1]                    
     map(--------------------------------------------------------------------
                                2                                    2       
         (Yy00000RE1*yRE1 - xRE1 , Yy00000RE1*xRE1 - yRE1, Yy00000RE1  - xRE1
                                                                             
     ------------------------------------------------------------------------
                  QQ[Yy , Yy , Yy ]
                       0    1    2
     -,---------------------------------------,{yRE1, xRE1, Yy00000RE1})}
          2                        2
     ) (Yy  - Yy , Yy Yy  - Yy , Yy  - Yy Yy )
          2     1    1  2     0    1     0  2

o2 : List
i3 : L#0

                QQ[Yy , Yy , Yy ]
                     0    1    2
o3 = ---------------------------------------
        2                        2
     (Yy  - Yy , Yy Yy  - Yy , Yy  - Yy Yy )
        2     1    1  2     0    1     0  2

o3 : QuotientRing

The previous example seminormalized a non-seminormal ring. Let’s try a seminormal ring.

i4 : R = QQ[x,y,z]/ideal(x^2*y-z^2);
i5 : L = seminormalize(R)

      QQ[Yy , Yy , Yy ]      QQ[Yy , Yy , Yy ]                    
           0    1    2            0    1    2                     
o5 = {-----------------, map(-----------------,R,{Yy , Yy , Yy }),
           2        2             2        2        1    2    0   
         Yy Yy  - Yy            Yy Yy  - Yy                       
           1  2     0             1  2     0                      
     ------------------------------------------------------------------------
                                                                             
                              QQ[Yy00000RE1, xRE1, yRE1, zRE1]               
     map(--------------------------------------------------------------------
                                                                         2   
         (Yy00000RE1*zRE1 - xRE1*yRE1, Yy00000RE1*xRE1 - zRE1, Yy00000RE1  - 
                                                                             
     ------------------------------------------------------------------------
           QQ[Yy , Yy , Yy ]
                0    1    2
     -----,-----------------,{zRE1, xRE1, yRE1})}
                2        2
     yRE1)    Yy Yy  - Yy
                1  2     0

o5 : List
i6 : L#0

     QQ[Yy , Yy , Yy ]
          0    1    2
o6 = -----------------
          2        2
        Yy Yy  - Yy
          1  2     0

o6 : QuotientRing

Ways to use seminormalize :