subquotient

subquotient(f,g) -- given matrices f and g with the same target, produces a new module representing the image of f in the cokernel of g.

The columns of f are called the generators, and the columns of g are the relations.

Functions:

generators -- recover the generators

relations -- recover the relations

prune -- convert to a module with presentation

This is the general form in which modules are represented, and subquotient modules are often returned as values of computations.

     i1 = R = ZZ/101[a..d]
     
     o1 = R
     
     o1 : PolynomialRing

     i2 = M = kernel vars R ++ cokernel vars R
     
     o2 = subquotient(| 0  0  0  -b -c -d 0 |,| 0 0 0 0 |)
                      | 0  -c -d a  0  0  0 | | 0 0 0 0 |
                      | -d b  0  0  a  0  0 | | 0 0 0 0 |
                      | c  0  b  0  0  a  0 | | 0 0 0 0 |
                      | 0  0  0  0  0  0  1 | | a b c d |
     
                                      5
          R - module, subquotient of R

     i3 = generators M
     
     o3 = | 0  0  0  -b -c -d 0 |
          | 0  -c -d a  0  0  0 |
          | -d b  0  0  a  0  0 |
          | c  0  b  0  0  a  0 |
          | 0  0  0  0  0  0  1 |
     
                  5       7
     o3 : Matrix R  <--- R

     i4 = relations M
     
     o4 = | 0 0 0 0 |
          | 0 0 0 0 |
          | 0 0 0 0 |
          | 0 0 0 0 |
          | a b c d |
     
                  5       4
     o4 : Matrix R  <--- R

     i5 = prune M
     
     o5 = cokernel | 0 0 0 0 0  0  -b -c |
                   | 0 0 0 0 0  -b 0  d  |
                   | 0 0 0 0 0  c  d  0  |
                   | 0 0 0 0 -c 0  a  0  |
                   | 0 0 0 0 d  a  0  0  |
                   | 0 0 0 0 b  0  0  a  |
                   | d c b a 0  0  0  0  |
     
                                   7
          R - module, quotient of R