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DGQuotientModule ** Ring -- Base change of a DG quotient module along a ring map

Description

Internally, the submodule is rebuilt directly inside the base-changed ambient (rather than constructing Sub ** S separately and then quotienting); this is what ensures the ambient DGModule of the result is the cached M ** S, not a distinct isomorphic copy.

i1 : R = ZZ/101[x,y]

o1 = R

o1 : PolynomialRing
 
i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o2 : DGAlgebra
 
i3 : Anat = A.natural

o3 = Anat

o3 : PolynomialRing, 2 skew commutative variable(s)
 
i4 : M = freeDGModule(A, {0})

o4 = {Base ring => R               }
      DG algebra => Anat
                            1
      Natural module => Anat
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o4 : DGModule
 
i5 : Sub = dgSubmodule(M, matrix {{x_Anat}})

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}}
                     1
     natural  => Anat
     inclusion => | x |

o5 : DGSubmodule
 
i6 : Q = M / Sub

o6 = DGQuotientModule Q = M / S
     Q.natural = cokernel | x |
     Degrees   = {{0, 0}}

o6 : DGQuotientModule
 
i7 : QR = R / ideal(x^2)

o7 = QR

o7 : QuotientRing
 
i8 : QS = Q ** QR

o8 = DGQuotientModule Q = M / S
     Q.natural = cokernel | x |
     Degrees   = {{0, 0}}

o8 : DGQuotientModule
 
i9 : isWellDefined QS

o9 = true
 
i10 : QS.ambient === M ** QR

o10 = true

See also

Ways to use this method:

The source of this document is in DGAlgebras/doc.m2:1896:0.