Macaulay2 » Documentation
Packages » Macaulay2Doc » A first course in commutative algebra » M2SingularBook » Singular Book 2.1.10
next | previous | forward | backward | up | index | toc

# Singular Book 2.1.10 -- submodules of A^n

A common method of creating a submodule of A^n in Macaulay2 is to take the image of a matrix. This will be a submodule generated by the columns of the matrix.

    i1 : A = QQ[x,y,z];

    i2 : f = matrix{{x*y-1,y^4},{z^2+3,x^3},{x*y*z,z^2}}

    o2 = | xy-1 y4 |
         | z2+3 x3 |
         | xyz  z2 |

                 3      2
    o2 : Matrix A  <-- A

    i3 : M = image f

    o3 = image | xy-1 y4 |
               | z2+3 x3 |
               | xyz  z2 |

                                 3
    o3 : A-module, submodule of A

    i4 : numgens M

    o4 = 2

    i5 : ambient M

          3
    o5 = A

    o5 : A-module, free

A submodule can easily be moved to quotient rings.

    i6 : Q = A/(x^2+y^2+z^2);

    i7 : substitute(M,Q)

    o7 = image | xy-1 y4       |
               | z2+3 -xy2-xz2 |
               | xyz  z2       |

                                 3
    o7 : Q-module, submodule of Q

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/M2-Singular-Book.m2:807:0.