homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map

Function: homology
Usage:

h = homology(f, n)
Inputs:
- f, an instance of the type DGModuleMap, A map M -> N of DG modules over a common DG algebra A.
- n, an integer, A homological degree.
Outputs:
- h, a matrix, The induced map H_n(f) : H_n(M) -> H_n(N), presented as a morphism of A.ring-modules.

Description

Computed via inducedMap from the chain map toComplexMap(f, n) applied to the pair of per-degree homology modules. Since a chain map sends cycles to cycles and boundaries to boundaries, the induced map is well defined.

The identity DG module map induces the identity on each H_n; the zero DG module map induces the zero map on each H_n. Composition respects homology: homology(g * f, n) == homology(g, n) * homology(f, n).

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 : k = R^1 / ideal(x, y)

o3 = cokernel | x y |

                            1
o3 : R-module, quotient of R

i4 : M = minimalSemifreeResolution(A, k, EndDegree => 3)

o4 = {Base ring => R                                                                                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       6
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}, {4, 4}, {4, 4}, {4, 4}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |, |     0    |, |     0    |, |     0    |}
                                   |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |  |     0    |
                                   |     0    |  |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |

o4 : DGModule

i5 : idM = identityDGModuleMap M

                               6
o5 = {Source => (R[T   ..T   ])        }
                    1,1   1,2
                               6
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => {0, 0} | 1 0 0 0 0 0 |
                 {2, 2} | 0 1 0 0 0 0 |
                 {2, 2} | 0 0 1 0 0 0 |
                 {4, 4} | 0 0 0 1 0 0 |
                 {4, 4} | 0 0 0 0 1 0 |
                 {4, 4} | 0 0 0 0 0 1 |

o5 : DGModuleMap

i6 : zM = zeroDGModuleMap(M, M)

                               6
o6 = {Source => (R[T   ..T   ]) }
                    1,1   1,2
                               6
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => 0

o6 : DGModuleMap

i7 : h0id = homology(idM, 0)

o7 = | 1 |

o7 : Matrix k <-- k

i8 : h0id == id_(source h0id)

o8 = true

i9 : h0z = homology(zM, 0)

o9 = 0

o9 : Matrix k <-- k

i10 : h0z == map(target h0z, source h0z, 0)

o10 = true

Multiplication by x on a minimal semifree resolution of k = R/(x, y) induces the zero map on H_0 = k, because x kills k.

i11 : natGens = apply(rank M.natural, i -> (M.natural)_i)

o11 = {| 1 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |}
       | 0 |  | 1 |  | 0 |  | 0 |  | 0 |  | 0 |
       | 0 |  | 0 |  | 1 |  | 0 |  | 0 |  | 0 |
       | 0 |  | 0 |  | 0 |  | 1 |  | 0 |  | 0 |
       | 0 |  | 0 |  | 0 |  | 0 |  | 1 |  | 0 |
       | 0 |  | 0 |  | 0 |  | 0 |  | 0 |  | 1 |

o11 : List

i12 : xMap = dgModuleMap(M, M, apply(natGens, g -> x * g))

                                6
o12 = {Source => (R[T   ..T   ])        }
                     1,1   1,2
                                6
       Target => (R[T   ..T   ])
                     1,1   1,2
       Natural => {0, 0} | x 0 0 0 0 0 |
                  {2, 2} | 0 x 0 0 0 0 |
                  {2, 2} | 0 0 x 0 0 0 |
                  {4, 4} | 0 0 0 x 0 0 |
                  {4, 4} | 0 0 0 0 x 0 |
                  {4, 4} | 0 0 0 0 0 x |

o12 : DGModuleMap

i13 : h0x = homology(xMap, 0)

o13 = 0

o13 : Matrix k <-- k

i14 : h0x == map(target h0x, source h0x, 0)

o14 = true

Ways to use this method:

homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4354:0.

homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map

Description

See also

Ways to use this method: