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HH DGQuotientModule -- The graded homology of a DG quotient module

Description

Implemented by taking toComplex and assembling the per-degree homologies HH_i(toComplex Q) into a direct sum, then tensoring each piece with HH(A) along A.ring -> HH(A). The two names homology Q and homologyModule Q are aliases, matching HH DGModule.

Because Q.natural is a cokernel rather than a free A.natural-module, the general cycle-action construction used for homologyModule(DGModule) does not apply; the result captures the R-module structure and the HH(A) action coming from A.ring -> HH(A) only.

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing
 
i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra
 
i3 : M = freeDGModule(A, {0, 1})

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

o3 : DGModule
 
i4 : natGens = apply(rank M.natural, i -> (M.natural)_i)

o4 = {| 1 |, | 0 |}
      | 0 |  | 1 |

o4 : List
 
i5 : setDiff(M, {0, x * natGens#0})

o5 = {Base ring => R                       }
      DG algebra => R[T   ]
                       1,1
                                 2
      Natural module => (R[T   ])
                            1,1
      Generator degrees => {{0, 0}, {1, 0}}
      Differentials on gens => {0, | x |}
                                   | 0 |

o5 : DGModule
 
i6 : S = dgSubmodule(M, (id_(M.natural))_{1})

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{1, 0}, {0, 0}}
                          2
     natural  => (R[T   ])
                     1,1
     inclusion => {0, 0} | 0 x |
                  {1, 0} | 1 0 |

o6 : DGSubmodule
 
i7 : Q = M / S

o7 = DGQuotientModule Q = M / S
     Q.natural = cokernel {0, 0} | 0 x |
                          {1, 0} | 1 0 |
     Degrees   = {{0, 0}, {1, 0}}

o7 : DGQuotientModule
 
i8 : HQ = homology Q

       ZZ 2
o8 = (---)
      101

      ZZ
o8 : ----module, free
     101
 
i9 : ring HQ === HH A

o9 = true
 
i10 : HQ' = homologyModule Q

        ZZ 2
o10 = (---)
       101

       ZZ
o10 : ----module, free
      101
 
i11 : HQ == HQ'

o11 = true

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4454:0.