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Module / Module -- quotient module

Synopsis

Description

If N is an ideal, ring element, or list or sequence of ring elements (in the ring of M), then the quotient is by the submodule N*M of M.

If N is a submodule of M, or a list or sequence of submodules, or a vector, then the quotient is by these elements or submodules.

i1 : R = ZZ/173[a..d]

o1 = R

o1 : PolynomialRing
i2 : M = ker matrix{{a^3-a*c*d,a*b*c-b^3,a*b*d-b*c^2}}

o2 = image {3} | 0      b3-abc bc2-abd |
           {3} | -c2+ad a3-acd 0       |
           {3} | b2-ac  0      a3-acd  |

                             3
o2 : R-module, submodule of R
i3 : M/a == M/(a*M)

o3 = true
i4 : M/M_0

o4 = subquotient ({3} | 0      b3-abc bc2-abd |, {3} | 0      |)
                  {3} | -c2+ad a3-acd 0       |  {3} | -c2+ad |
                  {3} | b2-ac  0      a3-acd  |  {3} | b2-ac  |

                               3
o4 : R-module, subquotient of R
i5 : M/(R*M_0 + b*M)

o5 = subquotient ({3} | 0      b3-abc bc2-abd |, {3} | 0      0        b4-ab2c  b2c2-ab2d |)
                  {3} | -c2+ad a3-acd 0       |  {3} | -c2+ad -bc2+abd a3b-abcd 0         |
                  {3} | b2-ac  0      a3-acd  |  {3} | b2-ac  b3-abc   0        a3b-abcd  |

                               3
o5 : R-module, subquotient of R
i6 : M/(M_0,a*M_1+M_2)

o6 = subquotient ({3} | 0      b3-abc bc2-abd |, {3} | 0      ab3-a2bc+bc2-abd |)
                  {3} | -c2+ad a3-acd 0       |  {3} | -c2+ad a4-a2cd          |
                  {3} | b2-ac  0      a3-acd  |  {3} | b2-ac  a3-acd           |

                               3
o6 : R-module, subquotient of R
i7 : presentation oo

o7 = {5} | -1 0  -a3+acd |
     {6} | 0  -a -c2+ad  |
     {6} | 0  -1 b2-ac   |

             3       3
o7 : Matrix R  <--- R

See also