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Macaulay2Doc :: Module ** Module

Module ** Module -- tensor product

Synopsis

Description

If M has generators m1, m2, ..., mr, and N has generators n1, n2, ..., ns, then M ** N has generators: m1**n1, m1**n2, ..., m2**n1, ..., mr**ns.
i1 : R = ZZ[a..d];
i2 : M = image matrix {{a,b}}

o2 = image | a b |

                             1
o2 : R-module, submodule of R
i3 : N = image matrix {{c,d}}

o3 = image | c d |

                             1
o3 : R-module, submodule of R
i4 : M ** N

o4 = cokernel {2} | -d 0  -b 0  |
              {2} | c  0  0  -b |
              {2} | 0  -d a  0  |
              {2} | 0  c  0  a  |

                            4
o4 : R-module, quotient of R
i5 : N ** M

o5 = cokernel {2} | -b 0  -d 0  |
              {2} | a  0  0  -d |
              {2} | 0  -b c  0  |
              {2} | 0  a  0  c  |

                            4
o5 : R-module, quotient of R

Use trim or minimalPresentation if a more compact presentation is desired.

Use flip(Module,Module) to produce the isomorphism M ** N --> N ** M.

To recover the factors from the tensor product, use the function formation.

See also