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toComplex(DGQuotientModule) -- Export a DG quotient module to a Complex

Description

For a DG quotient module Q = M / S, toComplex Q builds the cokernel at the Complex level: it exports the ambient DG module M via toComplex(DGModule), exports the inclusion of S via toComplexMap(DGModuleMap), and takes the cokernel of the resulting ComplexMap. If Q was previously passed through prune(DGQuotientModule) the cached pruned complex is returned instead.

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})

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

o3 : DGModule
 
i4 : z = zeroDGModuleMap(M, M)

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

o4 : DGModuleMap
 
i5 : Q = cokernel z

o5 = DGQuotientModule Q = M / S
                          1
     Q.natural = (R[T   ])
                     1,1
     Degrees   = {{0, 0}}

o5 : DGQuotientModule
 
i6 : CM = toComplex M

      1      1
o6 = R  <-- R
             
     0      1

o6 : Complex
 
i7 : CQ = toComplex Q

      1      1
o7 = R  <-- R
             
     0      1

o7 : Complex
 
i8 : assert(rank CM_0 == rank CQ_0)
 
i9 : assert(rank CM_1 == rank CQ_1)

See also

Ways to use this method:

The source of this document is in DGAlgebras/doc.m2:7841:0.