HH DGModuleMap -- The induced map on graded homology modules over HH(A)

Function: homology
Usage:

h = homology f
Inputs:
- f, an instance of the type DGModuleMap, A morphism M -> N of DG modules over a common DG algebra A. Both M.natural and N.natural must be free (the free / semifree case).
Outputs:
- h, The induced A.homology-module map H_*(M) -> H_*(N), where H_*(M) and H_*(N) carry their natural HH(A)-module structures via the action of cycles of A on M and N. The result lives over the ring HH(A).

Description

The per-degree restrictions of h are given by homology(DGModuleMap,ZZ). The present method assembles them into a single map of HH(A)-modules by forming the direct sum of the per-degree pieces and imposing the cycle-action relations on both sides. Functorial in f.

i1 : R = QQ[x,y]/ideal(x^2, y^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o2 : DGAlgebra

i3 : Mdg = minimalSemifreeResolution(A, R^1 / ideal(x,y), EndDegree => 2)

o3 = {Base ring => R                                          }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       3
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
                                   |     0    |  |     0    |
                                   |     0    |  |     0    |

o3 : DGModule

i4 : idM = identityDGModuleMap Mdg

                               3
o4 = {Source => (R[T   ..T   ])  }
                    1,1   1,2
                               3
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => {0, 0} | 1 0 0 |
                 {2, 2} | 0 1 0 |
                 {2, 2} | 0 0 1 |

o4 : DGModuleMap

i5 : h = homology idM
Finding easy relations           :  -- used 0.0124457s (cpu); 0.0112582s (thread); 0s (gc)

o5 = {0, 0} | 1 0 0 0 |
     {3, 4} | 0 1 0 0 |
     {3, 4} | 0 0 1 0 |
     {3, 4} | 0 0 0 1 |

o5 : Matrix

i6 : ring source h === HH A

o6 = true

Ways to use this method:

HH DGModuleMap -- The induced map on graded homology modules over HH(A)

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:515:0.

HH DGModuleMap -- The induced map on graded homology modules over HH(A)

Description

See also

Ways to use this method: