Macaulay2 » Documentation
Packages » DGAlgebras » Homology of DG modules and DG module maps » HH DGModule
next | previous | forward | backward | up | index | toc

HH DGModule -- The graded homology of a DG module as a module over HH(A)

Description

The two names homology M and homologyModule M produce the same object and are provided as aliases, matching the analogous convention for DG algebras in which HH DGAlgebra returns homologyAlgebra.

For per-degree homology as an A.ring-module, use HH_ZZ DGModule.

A fast path is taken when M came from koszulComplexDGM and has vanishing generator differentials: in that case the result is computed directly from M.module via homologyModule(DGAlgebra,Module). Otherwise the method builds the HH(A)-module presentation degree by degree, using the action of each cycle class of A on M.

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 => 2)

o4 = {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    |

o4 : DGModule
 
i5 : HM = homology M
Finding easy relations           :  -- used 0.0121349s (cpu); 0.011128s (thread); 0s (gc)

o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
              {3, 4} | 0   0   X_1 X_2 0   0   |
              {3, 4} | 0   0   0   X_1 0   X_2 |
              {3, 4} | 0   0   0   0   X_2 X_1 |

                                                4
o5 : QQ[X ..X ]-module, quotient of (QQ[X ..X ])
         1   2                           1   2
 
i6 : ring HM === HH A

o6 = true
 
i7 : HM' = homologyModule M

o7 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
              {3, 4} | 0   0   X_1 X_2 0   0   |
              {3, 4} | 0   0   0   X_1 0   X_2 |
              {3, 4} | 0   0   0   0   X_2 X_1 |

                                                4
o7 : QQ[X ..X ]-module, quotient of (QQ[X ..X ])
         1   2                           1   2
 
i8 : HM == HM'

o8 = true

The ring of the result is HH(A), the homology algebra of the underlying DG algebra.

Caveat

The general path currently requires M.natural to be a free A.natural-module (or M to be a koszulComplexDGM with zero generator differentials).

See also

Ways to use this method:

  • HH DGModule -- The graded homology of a DG module as a module over HH(A)

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4299:0.