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Macaulay2Doc :: tensor(Ring,RingMap,Matrix)

tensor(Ring,RingMap,Matrix) -- tensor product via a ring map

Synopsis

Description

None of the options are relevant for these uses of tensor.

i1 : R = QQ[a..d]

o1 = R

o1 : PolynomialRing
i2 : S = QQ[s,t]

o2 = S

o2 : PolynomialRing
i3 : F = map(S,R,{s^4,s^3*t,s*t^3,t^4})

               4   3      3   4
o3 = map(S,R,{s , s t, s*t , t })

o3 : RingMap S <--- R
i4 : f = matrix{{a,b,c,d}}

o4 = | a b c d |

             1       4
o4 : Matrix R  <--- R
i5 : tensor(F,f)

o5 = | s4 s3t st3 t4 |

             1       4
o5 : Matrix S  <--- S
i6 : tensor(F,image f)

o6 = cokernel {1} | -s3t 0    -st3 0   0   -t4 |
              {1} | s4   -st3 0    0   -t4 0   |
              {1} | 0    s3t  s4   -t4 0   0   |
              {1} | 0    0    0    st3 s3t s4  |

                            4
o6 : S-module, quotient of S

If the ring S is given as an argument, then it must match the target of F, and the result is identical to the version without S given. The reason it is here is to mimic natural mathematical notation: S **_R M.

i7 : tensor(S,F,f)

o7 = | s4 s3t st3 t4 |

             1       4
o7 : Matrix S  <--- S
i8 : tensor(S,F,image f)

o8 = cokernel {1} | -s3t 0    -st3 0   0   -t4 |
              {1} | s4   -st3 0    0   -t4 0   |
              {1} | 0    s3t  s4   -t4 0   0   |
              {1} | 0    0    0    st3 s3t s4  |

                            4
o8 : S-module, quotient of S