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

Usage:

phiTilde = liftToDGMap(B,A,phi)
Inputs:
- B, an instance of the type DGAlgebra, Target
- A, an instance of the type DGAlgebra, Source
- phi, a ring map, Map from A in degree zero to B in degree zero
Optional inputs:
- EndDegree => ..., default value -1
Outputs:
- phiTilde, an instance of the type DGAlgebraMap, DGAlgebraMap lifting phi to a map of DGAlgebras.

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   ]
                               1,1   2,3
                                 2       2       2
      Differential => {a, b, c, a T   , b T   , c T   }
                                   1,1     1,2     1,3

o4 : DGAlgebra

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

o5 = {Ring => S                                                                                                                                                                    }
      Underlying algebra => S[T   ..T   , T   ..T   , T   ..T   , T   ..T   ]
                               1,1   1,3   2,1   2,4   3,1   3,3   4,1   4,6
                                 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   }
                                   1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4

o5 : DGAlgebra

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

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

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 : ComplexMap

Ways to use liftToDGMap:

liftToDGMap(DGAlgebra,DGAlgebra,RingMap)

For the programmer

The object liftToDGMap is a method function with options.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:9093:0.