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DGAlgebras :: torMap

torMap -- Compute the map of Tor algebras associated to a RingMap.

Synopsis

Description

The functor TorR(M,N) is also functorial in the ring argument. Therefore, a ring map phi from A to B induces an algebra map from the Tor algebra of A to the Tor algebra of B.

i1 : R = ZZ/101[a,b,c]/ideal{a^3,b^3,c^3,a^2*b^2*c^2}

o1 = R

o1 : QuotientRing
i2 : S = R/ideal{a*b^2*c^2,a^2*b*c^2,a^2*b^2*c}

o2 = S

o2 : QuotientRing
i3 : f = map(S,R)

o3 = map(S,R,{a, b, c})

o3 : RingMap S <--- R
i4 : fTor = torMap(f,GenDegreeLimit=>3)

          ZZ                                                                              ZZ
o4 = map(---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  ],---[X , X , X , X , X , X , X , X , X , X  ],{X , X , X , X , X , X , 0, 0, 0, 0})
         101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17  101  1   2   3   4   5   6   7   8   9   10    1   2   3   4   5   6

              ZZ                                                                                   ZZ
o4 : RingMap ---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  ] <--- ---[X , X , X , X , X , X , X , X , X , X  ]
             101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17       101  1   2   3   4   5   6   7   8   9   10
i5 : matrix fTor

o5 = | X_1 X_2 X_3 X_4 X_5 X_6 0 0 0 0 |

              ZZ                                                                             1        ZZ                                                                             10
o5 : Matrix (---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  ])  <--- (---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  ])
             101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17          101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17

In the following example, the map on Tor is surjective, which means that the ring homomorphism is large (Dress-Kramer).

i6 : R = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d}

o6 = R

o6 : QuotientRing
i7 : S = ZZ/101[a,b]/ideal{a^3,b^3}

o7 = S

o7 : QuotientRing
i8 : f = map(S,R,matrix{{a,b,0,0}})

o8 = map(S,R,{a, b, 0, 0})

o8 : RingMap S <--- R
i9 : fTor = torMap(f,GenDegreeLimit=>4)

          ZZ                  ZZ
o9 = map(---[X , X , X , X ],---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  ],{X , X , 0, 0, 0, 0, 0, 0, X , X , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0})
         101  1   2   3   4  101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50   51   52   53   54   55    1   2                     3   4

              ZZ                       ZZ
o9 : RingMap ---[X , X , X , X ] <--- ---[X , X , X , X , X , X , X , X , X , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  , X  ]
             101  1   2   3   4       101  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50   51   52   53   54   55
i10 : matrix fTor

o10 = | X_1 X_2 0 0 0 0 0 0 X_3 X_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

               ZZ                 1        ZZ                 55
o10 : Matrix (---[X , X , X , X ])  <--- (---[X , X , X , X ])
              101  1   2   3   4          101  1   2   3   4

Ways to use torMap :