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DGAlgebras :: liftToDGMap

liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism

Synopsis

Description

In order for phiTilde to be defined, phi of the image of the differential of A in degree 1 must lie in the image of the differential of B in degree 1. At present, this condition is not checked.

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

o1 = R

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

o2 = S

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

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

o3 : RingMap S <--- R
i4 : A = acyclicClosure(R,EndDegree=>3)

o4 = {Ring => R                                  }
      Underlying algebra => R[T ..T ]
                               1   6
                                 2     2     2
      Differential => {a, b, c, a T , b T , c T }
                                   1     2     3

o4 : DGAlgebra
i5 : B = acyclicClosure(S,EndDegree=>3)

o5 = {Ring => S                                                                                                                              }
      Underlying algebra => S[T ..T  ]
                               1   16
                                 2     2     2       2 2     2 2      2 2      2 2     2 2        2 2       2 2        2       2       2
      Differential => {a, b, c, a T , b T , c T , a*b c T , b c T , -a b T , -a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
                                   1     2     3         1       4        6        5       3 4        3 5       2 4      1 7     3 7     2 7

o5 : DGAlgebra
i6 : phi = liftToDGMap(B,A,f)

o6 = map (S[T ..T  ], R[T ..T ], {T , T , T , T , T , T , a, b, c})
             1   16      1   6     1   2   3   4   5   6

o6 : DGAlgebraMap
i7 : toComplexMap(phi,EndDegree=>3)

                                         1
o7 = 0 : cokernel | a2b2c2 | <--------- R  : 0
                                | 1 |

                                                                   3
     1 : cokernel {1} | a2b2c2 0      0      | <----------------- R  : 1
                  {1} | 0      a2b2c2 0      |    {1} | 1 0 0 |
                  {1} | 0      0      a2b2c2 |    {1} | 0 1 0 |
                                                  {1} | 0 0 1 |

                                                                                                     6
     2 : cokernel {2} | a2b2c2 0      0      0      0      0      0      | <----------------------- R  : 2
                  {2} | 0      a2b2c2 0      0      0      0      0      |    {2} | 1 0 0 0 0 0 |
                  {2} | 0      0      a2b2c2 0      0      0      0      |    {2} | 0 1 0 0 0 0 |
                  {3} | 0      0      0      a2b2c2 0      0      0      |    {2} | 0 0 1 0 0 0 |
                  {3} | 0      0      0      0      a2b2c2 0      0      |    {3} | 0 0 0 1 0 0 |
                  {3} | 0      0      0      0      0      a2b2c2 0      |    {3} | 0 0 0 0 1 0 |
                  {6} | 0      0      0      0      0      0      a2b2c2 |    {3} | 0 0 0 0 0 1 |
                                                                              {6} | 0 0 0 0 0 0 |

                                                                                                                                                                            10
     3 : cokernel {3} | a2b2c2 0      0      0      0      0      0      0      0      0      0      0      0      0      0      0      | <------------------------------- R   : 3
                  {4} | 0      a2b2c2 0      0      0      0      0      0      0      0      0      0      0      0      0      0      |    {3} | 1 0 0 0 0 0 0 0 0 0 |
                  {4} | 0      0      a2b2c2 0      0      0      0      0      0      0      0      0      0      0      0      0      |    {4} | 0 1 0 0 0 0 0 0 0 0 |
                  {4} | 0      0      0      a2b2c2 0      0      0      0      0      0      0      0      0      0      0      0      |    {4} | 0 0 1 0 0 0 0 0 0 0 |
                  {4} | 0      0      0      0      a2b2c2 0      0      0      0      0      0      0      0      0      0      0      |    {4} | 0 0 0 1 0 0 0 0 0 0 |
                  {4} | 0      0      0      0      0      a2b2c2 0      0      0      0      0      0      0      0      0      0      |    {4} | 0 0 0 0 1 0 0 0 0 0 |
                  {4} | 0      0      0      0      0      0      a2b2c2 0      0      0      0      0      0      0      0      0      |    {4} | 0 0 0 0 0 1 0 0 0 0 |
                  {4} | 0      0      0      0      0      0      0      a2b2c2 0      0      0      0      0      0      0      0      |    {4} | 0 0 0 0 0 0 1 0 0 0 |
                  {4} | 0      0      0      0      0      0      0      0      a2b2c2 0      0      0      0      0      0      0      |    {4} | 0 0 0 0 0 0 0 1 0 0 |
                  {4} | 0      0      0      0      0      0      0      0      0      a2b2c2 0      0      0      0      0      0      |    {4} | 0 0 0 0 0 0 0 0 1 0 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      a2b2c2 0      0      0      0      0      |    {4} | 0 0 0 0 0 0 0 0 0 1 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      0      a2b2c2 0      0      0      0      |    {7} | 0 0 0 0 0 0 0 0 0 0 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      0      0      a2b2c2 0      0      0      |    {7} | 0 0 0 0 0 0 0 0 0 0 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      0      0      0      a2b2c2 0      0      |    {7} | 0 0 0 0 0 0 0 0 0 0 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      0      0      0      0      a2b2c2 0      |    {7} | 0 0 0 0 0 0 0 0 0 0 |
                  {7} | 0      0      0      0      0      0      0      0      0      0      0      0      0      0      0      a2b2c2 |    {7} | 0 0 0 0 0 0 0 0 0 0 |
                                                                                                                                             {7} | 0 0 0 0 0 0 0 0 0 0 |

o7 : ChainComplexMap

Ways to use liftToDGMap :

For the programmer

The object liftToDGMap is a method function with options.