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

Synopsis

Usage:

torPhi = torMap(phi)
Inputs:
- phi, a ring map,
Optional inputs:
- GenDegreeLimit => ..., default value 3
Outputs:
- torPhi, a ring map,

Description

The functor Tor_R(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 , 0, 0, 0, 0})
          101  1   17   101  1   10     1   2   3   4   5   6

              ZZ               ZZ
o4 : RingMap ---[X ..X  ] <-- ---[X ..X  ]
             101  1   17      101  1   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  ])
             101  1   17         101  1   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 , 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   4   101  1   55     1   2                     3   4

              ZZ              ZZ
o9 : RingMap ---[X ..X ] <-- ---[X ..X  ]
             101  1   4      101  1   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 ])
              101  1   4         101  1   4

Ways to use torMap :

torMap(RingMap)

For the programmer

The object torMap is a method function with options.