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Macaulay2Doc :: ambient(Module)

ambient(Module) -- ambient free module

Synopsis

Description

If a module is a submodule or quotient of a free module F, or is a subquotient of F (that is, a submodule of a quotient of F), then this routine yields the free module F.
i1 : R = QQ[x_1 .. x_5]

o1 = R

o1 : PolynomialRing
i2 : N = image matrix{{x_1,x_2},{x_2,x_3}}

o2 = image | x_1 x_2 |
           | x_2 x_3 |

                             2
o2 : R-module, submodule of R
i3 : ambient N

      2
o3 = R

o3 : R-module, free
i4 : ambient cokernel vars R

      1
o4 = R

o4 : R-module, free
i5 : ambient kernel vars R

      5
o5 = R

o5 : R-module, free, degrees {5:1}
i6 : M = image vars R ++ cokernel vars R

o6 = subquotient (| x_1 x_2 x_3 x_4 x_5 0 |, | 0   0   0   0   0   |)
                  | 0   0   0   0   0   1 |  | x_1 x_2 x_3 x_4 x_5 |

                               2
o6 : R-module, subquotient of R
i7 : ambient M

      2
o7 = R

o7 : R-module, free
This module is always the common target free module of the generator and relation matrices of M
i8 : ambient M == target generators M

o8 = true
i9 : ambient M == target relations M

o9 = true

See also