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subquotient -- make a subquotient module

Synopsis

Description

If g is omitted, then the identity map on the target of r is used as g, and the cokernel of r is returned. If r is omitted (but not the comma), then a zero map to the target of g is used as r, and the image of g is returned. If both are omitted, then M is returned.

See subquotient modules for an overview of subquotient modules in Macaulay2.

The general form in which modules are represented in Macaulay 2 is as subquotients, and subquotient modules are often returned as values of computations, as in the example below.

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

o1 = R

o1 : PolynomialRing
i2 : M = kernel vars R ++ cokernel vars R

o2 = subquotient ({1} | -b 0  -c 0  0  -d 0 |, {1} | 0 0 0 0 |)
                  {1} | a  -c 0  0  -d 0  0 |  {1} | 0 0 0 0 |
                  {1} | 0  b  a  -d 0  0  0 |  {1} | 0 0 0 0 |
                  {1} | 0  0  0  c  b  a  0 |  {1} | 0 0 0 0 |
                  {0} | 0  0  0  0  0  0  1 |  {0} | a b c d |

                               5
o2 : R-module, subquotient of R
i3 : generators M

o3 = {1} | -b 0  -c 0  0  -d 0 |
     {1} | a  -c 0  0  -d 0  0 |
     {1} | 0  b  a  -d 0  0  0 |
     {1} | 0  0  0  c  b  a  0 |
     {0} | 0  0  0  0  0  0  1 |

             5       7
o3 : Matrix R  <--- R
i4 : relations M

o4 = {1} | 0 0 0 0 |
     {1} | 0 0 0 0 |
     {1} | 0 0 0 0 |
     {1} | 0 0 0 0 |
     {0} | a b c d |

             5       4
o4 : Matrix R  <--- R
i5 : M === subquotient(generators M, relations M)

o5 = true
i6 : prune M,

o6 = (cokernel {2} | 0 0 0 0 c  0  0  d  |, )
               {2} | 0 0 0 0 a  d  0  0  |
               {2} | 0 0 0 0 -b 0  d  0  |
               {2} | 0 0 0 0 0  b  a  0  |
               {2} | 0 0 0 0 0  -c 0  a  |
               {2} | 0 0 0 0 0  0  -c -b |
               {0} | d c b a 0  0  0  0  |

o6 : Sequence

See also

Ways to use subquotient :