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HH_ZZ DGQuotientModule -- The degree-n homology of a DG quotient module

Description

Implemented via HH_n (toComplex Q). Since the underlying natural module is a cokernel rather than a free module, the differentials of Q are expressed through toComplex and the standard M2 HH_n is applied.

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 : h0 = prune homology(0, Q)

o8 = cokernel | x 0 0 |

                            1
o8 : R-module, quotient of R
 
i9 : h1 = prune homology(1, Q)

o9 = cokernel | x 0 |

                            1
o9 : R-module, quotient of R
 
i10 : h2 = prune homology(2, Q)

o10 = 0

o10 : R-module
 
i11 : numgens h0 == 1

o11 = true
 
i12 : numgens h1 == 1

o12 = true
 
i13 : h2 == 0

o13 = true

With S acyclic and H_0(M) = R/x, the quotient Q = M/S has H_0(Q) = R/x and H_1(Q) = R/x, both equal to the residue field k = ZZ/101 (one generator each, torsion), and H_2(Q) = 0.

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.m2:4404:0.