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codim(Ideal) -- compute the codimension

Synopsis

Usage:
codim I
Function: codim
Inputs:
- I, an ideal
Outputs:
- an integer

Description

Computes the codimension of the ideal I.

i1 : R = ZZ/101[a..e];

i2 : I = monomialCurveIdeal(R,{2,3,5,7})

             2                               2     2    3           3      2
o2 = ideal (d  - c*e, b*d - a*e, b*c - a*d, c d - b e, c  - a*b*e, b  - a*c )

o2 : Ideal of R

i3 : J = ideal presentation singularLocus(R/I)

             2                               2     2    3           3      2 
o3 = ideal (d  - c*e, b*d - a*e, b*c - a*d, c d - b e, c  - a*b*e, b  - a*c ,
     ------------------------------------------------------------------------
        2       2    3                        2      2                   
     - d e + c*e , 2d  - 2c*d*e, - b*d*e + a*e , 2b*d  - b*c*e - a*d*e, -
     ------------------------------------------------------------------------
        2    2                                                 2         
     c*d  + c e, b*c*e - a*d*e, - b*c*d + a*c*e, - b*c*d + 2a*d  - a*c*e,
     ------------------------------------------------------------------------
        2           2              2        3        2        2     2 2 
     b*c  - a*c*d, b c - 2a*b*d + a e, -2b*e , 4b*d*e , - 4c*d e - c e ,
     ------------------------------------------------------------------------
         3    2        2 2         2    2       2 2      2 2    2     
     4c*d  + c d*e + 2b e , -2b*c*e , 2c d*e + b e , - 2c d  - b d*e -
     ------------------------------------------------------------------------
           2   3      2        3      2 2     2                    2   
     2a*b*e , c e - 2b d*e, - c d + 2b d  - 2b c*e + 4a*b*d*e, 2b*c d -
     ------------------------------------------------------------------------
           2    3       2          2      2         3    2       2 2    3   
     4a*c*d  + b e - a*c e, -2b*d*e , 4b*d e, - 4c*d  - c d*e, 4c d  + c e +
     ------------------------------------------------------------------------
       2            2               2 2    2         3      2      3   
     4b d*e - 2a*b*e , -2b*c*d*e, 2c d  + b d*e, - 2c d - 3b c*e, c d -
     ------------------------------------------------------------------------
       2 2     4     2                     3     3        2       2       3  
     2b d , - c  + 2b c*d + 2a*b*c*e, - b*c  + 2b d - 2a*c d - a*b e, 2c*d  -
     ------------------------------------------------------------------------
       2        2 2   2 2    3      2            2          2      2   
     2c d*e - 2b e , c d  - c e + 2b d*e + 2a*b*e , - 2b*c*d  - b*c e -
     ------------------------------------------------------------------------
                   2          2     3      2 2    2                      2  
     2a*c*d*e, 3b*c d + 2a*c*d  + 2b e, - b d  + b c*e - 2a*b*d*e, 2a*c*d  +
     ------------------------------------------------------------------------
      3      3        2        2      3       2       2    3       3      2 
     b e, - b d - 2a*c d - 2a*b e, - b d + a*c d - a*b e, b c - a*c  + a*b d
     ------------------------------------------------------------------------
         2      4        2     2            2      3      2          2   
     + 2a b*e, b  - a*b*c  - 2a c*d, - b*d*e  - a*e , 2b*d e + 2a*d*e , -
     ------------------------------------------------------------------------
       2       2 2    2 2        2                   2    3       3   
     6c d*e + b e , 6c d  + a*b*e , - b*c*d*e - a*c*e , 3c e, - 3c d -
     ------------------------------------------------------------------------
                2 2    2           2               2         3       2   
     a*b*d*e - a e , -b c*e, 2a*b*d  - a*b*c*e + 2a d*e, 3b*c  - 6a*c d +
     ------------------------------------------------------------------------
        2          2        2                 2       2 2     2           2 
     a*b e, - b*c*e  - a*d*e , 2b*c*d*e + 2a*d e, - 6c d  + 2b d*e - a*b*e ,
     ------------------------------------------------------------------------
       3                2 2       2               3     2                   4
     6c d + 2a*b*d*e - a e , - b*c e - a*c*d*e, 3c d - b c*e + a*b*d*e, - 3c 
     ------------------------------------------------------------------------
                         2                        2           3       2   
     - 2a*b*c*e, - 2a*b*d  + a*b*c*e, 2a*b*c*d + a c*e, - 3a*c  + 2a*b d -
     ------------------------------------------------------------------------
      2       2 2     3     2           2   2                  2 2        2 
     a b*e, 3c d  - 3c e - b d*e - a*b*e , b c*e + 2a*b*d*e + a e , - 3b*c d
     ------------------------------------------------------------------------
        3        2       3       2       2          2    2         2    
     + b e - 3a*c e, 3b*c  + 3a*c d + a*b e, - a*b*d  - a d*e, 3a*c d, -
     ------------------------------------------------------------------------
         3      2     2         2      2     2       3      3     2 2    2 3 
     3a*c  - a*b d - a b*e, -a*b d, a*b c + a b*d + a e, a*b  - 3a c , 2b e ,
     ------------------------------------------------------------------------
        2   2        2       2 2         2            2       2 2    2   2   
     -4b d*e , 4b*c*d e + b*c e , - 12b*c d*e + 4a*c*d e + a*c e , 2b c*e , -
     ------------------------------------------------------------------------
         2       3 2      3        2         2 2       3      3          3   
     2b*c d*e - b e , 6b*c e - 2a*c d*e + a*b e , - b*c e + 2b d*e, - a*c e -
     ------------------------------------------------------------------------
         2       5     2 2            2        2           2        2 2  
     2a*b d*e, 3c  - 6b c d + 4a*b*c*d  + a*b*c e, - 2b*c*d e + 6b*c e  -
     ------------------------------------------------------------------------
             2       2        3 2      2 2    4      2           4      2 2 
     2a*c*d*e , - b*c d*e - 2b e  - a*c e , 3c e + 2b c*d*e, - 3c d - 6b c e
     ------------------------------------------------------------------------
                    3         2 2     2 2     2 2            2         2   
     + 2a*b*c*d*e, b d*e + a*b e , -3b c e, 3b c d - 2a*b*c*d  + 6a*b*c e -
     ------------------------------------------------------------------------
       2        4          2       3     2 2       3 2       4       2    
     2a c*d*e, b e, - a*b*c d - a*b e - a c e, - 3b c  + 3a*c  + 2a*b c*d,
     ------------------------------------------------------------------------
         2            2      3 2       3       2          2 2    4     2 2   
     4b*c d*e - 2a*c*d e - 2b e , - b*c e - a*c d*e + 2a*b e , 3c d - b c e -
     ------------------------------------------------------------------------
                     5         2      2        3         2         2 2   
     2a*b*c*d*e, - 3c  + 5a*b*c e - 2a c*d*e, b c*e - a*b d*e, - 3b c d +
     ------------------------------------------------------------------------
             2    4     2 3         2       3        2       3          3  
     2a*b*c*d  + b e, 3b c  - 2a*b*c d - a*b e, a*b*c d - a*b e, - a*b*c  +
     ------------------------------------------------------------------------
      2 2       2 2     2          2        2 2    2 2     2        2   
     a b e, 2a*b c  - 2a b*c*d, - c d*e + 3b e , 2c d  - 6b d*e, b*c e +
     ------------------------------------------------------------------------
                       2     3      3      2          2       2        2    
     4a*c*d*e, - 4a*c*d  - 3b e, - c d + 3b c*e, -3a*c e, 2a*c d + 3a*b e, -
     ------------------------------------------------------------------------
        3       2     3        2           2     2        3      2       3   
     b*c  + 2a*c d, 3b c - 6a*b d, - 2a*b*c  + 4a c*d, - c e + 3b d*e, 2c d -
     ------------------------------------------------------------------------
       2 2      2          2      2       3        2        2      4  
     6b d , 2b*c d + 4a*c*d  - a*c e, - 6b d - 4a*c d + 3a*b e, - c  +
     ------------------------------------------------------------------------
       2          3       2     3        3     3       2     2 2       2   
     3b c*d, - b*c  - 2a*c d, 3b c + 2a*c , a*c , -3a*b c, 2a c , - b*c d -
     ------------------------------------------------------------------------
           2     3        2      3     3       2        2    2 2             
     2a*c*d  + 3b e + 2a*c e, b*c  - 3b d + a*c d - 3a*b e, b c  + 2a*b*c*d +
     ------------------------------------------------------------------------
       2         4         2     2          3       2          2     2    
     2a c*e, - 3b  - 2a*b*c  - 2a c*d, - a*c  + 3a*b d, - a*b*c  - 2a c*d,
     ------------------------------------------------------------------------
         3     2 2   2 2     2 2    3       2 2       2       3 2    4    
     3a*b  + 2a c , a c , -3a b , 2a c, 2b*c e , -4b*c d*e, 4c d  + c e, -
     ------------------------------------------------------------------------
        2   2     2 2                    3       4     2 2     2 2      4   
     12b c*d  - 3b c e + 8a*b*c*d*e, 2b*c e, - 2c d - b c e, 6b c d + 3b e -
     ------------------------------------------------------------------------
           2      5     2 2     2 3     4         4       2         3 2  
     4a*b*c e, - c  + 2b c d, 3b c  - 6b d, - 2a*c  + 4a*b c*d, - 2c d  +
     ------------------------------------------------------------------------
       2                2     4     2 2       3        3       3          3 
     6b c*d*e - 4a*b*c*e , - c d + b c e, 2b*c d - 2a*c e, - 6b c*d + 2a*c d
     ------------------------------------------------------------------------
           2      2 2      4          2         3        2         2     
     + 4a*b c*e, b c d - 3b e + 2a*b*c e, - 2a*c d + 2a*b c*e, 4a*b c*d -
     ------------------------------------------------------------------------
       2        3 2      4      5       2 2      3      2 3      4      2   2
     4a b*c*e, b c  - a*c , - 3b  + 3a*b c , 2a*b c - 2a c , - 2c d + 6b c*d 
     ------------------------------------------------------------------------
         2 2                   5     2 2          2        4       3     3 2
     - 2b c e - 4a*b*c*d*e, - c  + 3b c d + 2a*b*c e, - b*c  - 4a*c d, 3b c 
     ------------------------------------------------------------------------
           4       2        2        2 3     4    3 2       2         5  
     + 2a*c  + 6a*b c*d - 4a b*c*e, b c  - 3b d, b c  + 2a*b c*d, - 3b  -
     ------------------------------------------------------------------------
         2 2      2 2      4     2 2     3 2      2 2      3          2    
     2a*b c , -a*b c , 3a*b , -2a b c, 3b e  + a*c e , - 6b d*e - 2a*c d*e,
     ------------------------------------------------------------------------
       4                     2 2      2         3         3       5  
     6c d + 4a*b*c*d*e, - 18b c d + 4a c*d*e, 3b c*e + a*c e, - 3c  -
     ------------------------------------------------------------------------
           2     2 3       3      2 2         2        3     2           4   
     3a*b*c e, 9b c  + 3a*b e - 2a c e, 2a*b*c d, -6a*b d, 4a b*c*d, - 3c d +
     ------------------------------------------------------------------------
       2 2                   2   2      4         2       4       2      
     9b c e - 2a*b*c*d*e - 2a c*e , - 3b e - a*b*c e, 3b*c  + 2a*b c*e, -
     ------------------------------------------------------------------------
       3 2     2            2     2 2        4      2 2     2          3    
     9b c  + 2a b*c*e, a*b*c d + a c e, -3a*c , 9a*b c  - 2a b*c*d - 2a c*e,
     ------------------------------------------------------------------------
        2 2       4    2 2       5     2 2      4          2      2      
     a*b c , -3a*b , 2a b c, - 3c  + 9b c d - 3b e - 3a*b*c e - 2a c*d*e,
     ------------------------------------------------------------------------
         3     2 2         4     2           2 2     3          3       3  
     3a*b e + a c e, - 3a*c  - 2a b*c*e, 9a*b c  - 2a c*e, a*b*c  - 3a*b d,
     ------------------------------------------------------------------------
        2 2     2             4     2   2    2   2    2 3     3         4   
     a*b c  + 2a b*c*d, - 3a*b  - 2a b*c , -a b*c , 3a b , -2a b*c, 6b*c e -
     ------------------------------------------------------------------------
       3            3          2   2      3 2       4     6         3    
     6b c*d*e - 2a*c d*e + 4a*b c*e , - 3b c e - a*c e, 3c  + 2a*b*c e, -
     ------------------------------------------------------------------------
       2 4     2 3     5       2 2       2 4         3        3       4 2  
     9b c  + 2a c e, 3b e - a*b c e, - 3b c  + 2a*b*c d - 2a*b c*e, 9b c  -
     ------------------------------------------------------------------------
         3        2 2          4       3 2    2   3
     6a*b c*d + 2a b c*e, a*b*c , -3a*b c , 2a b*c )

o3 : Ideal of R

i4 : codim J

o4 = 4

i5 : radical J

o5 = ideal (d, c, b, a*e)

o5 : Ideal of R

Caveat

We don't really compute the codimension of I when the basering R of I is not equidimensional! What codim actually computes is dim(R)-dim(R/I).

See also

dim(Ideal) -- compute the Krull dimension
dim(MonomialIdeal) -- compute the Krull dimension