HH DGAlgebraMap -- The induced ring map on homology algebras

Function: homology
Usage:

homologyPhi = homology phi

homologyPhi = HH phi
Inputs:
- phi, an instance of the type DGAlgebraMap,
Outputs:
- homologyPhi, a ring map, the induced map HH(source phi) -> HH(target phi)

Description

A DGAlgebraMap phi: A -> B is a chain map, so it descends to a ring map on homology algebras. This function computes that induced map, returning a RingMap from homologyAlgebra of the source to homologyAlgebra of the target. The homology algebras are computed via homologyAlgebra if they have not been cached.

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

o1 = R

o1 : QuotientRing

i2 : K1 = koszulComplexDGA(ideal vars R, Variable => "Y")

o2 = {Ring => R                          }
      Underlying algebra => R[Y   ..Y   ]
                               1,1   1,3
      Differential => {a, b, c}

o2 : DGAlgebra

i3 : K2 = koszulComplexDGA(ideal{b, c}, Variable => "T")

o3 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {b, c}

o3 : DGAlgebra

i4 : g = dgAlgebraMap(K1, K2, matrix{{Y_(1,2), Y_(1,3)}})

o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
             1,1   1,3      1,1   1,2     1,2   1,3

o4 : DGAlgebraMap

i5 : HHg = HH g
Finding easy relations           :  -- used 0.0140702s (cpu); 0.013303s (thread); 0s (gc)

                         ZZ
                        ---[a..c]
           ZZ           101
o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
          101  1   2           3   1     1
                       (c, b, a )

                               ZZ
                              ---[a..c]
              ZZ              101
o5 : RingMap ---[X ..X ] <-- ----------[X ]
             101  1   2              3   1
                             (c, b, a )

i6 : source HHg === homologyAlgebra K1

o6 = false

i7 : target HHg === homologyAlgebra K2

o7 = false

Ways to use this method:

HH DGAlgebraMap -- The induced ring map on homology algebras

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:9127:0.

HH DGAlgebraMap -- The induced ring map on homology algebras

Description

See also

Ways to use this method: