DGSubmodule ** Ring -- Base change of a DG submodule along a ring map

Operator: **
Usage:

SubS = Sub ** S
Inputs:
- Sub, an instance of the type DGSubmodule, A DG submodule of some ambient DG module M.
- S, a ring, A ring admitting substitution from the ground ring of Sub.
Outputs:
- SubS, an instance of the type DGSubmodule, The DG submodule of M ** S whose inclusion matrix is obtained by substituting every entry of the original inclusion into the base-changed ring.

Description

The ambient is (Sub.ambient) ** S, so that for any DGQuotientModule built from Sub the quotient (M/Sub) ** S equals (M ** S) / (Sub ** S) up to canonical equality.

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, y_Anat}})

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}}
                     2
     natural  => Anat
     inclusion => | x y |

o5 : DGSubmodule

i6 : QR = R / ideal(x^2)

o6 = QR

o6 : QuotientRing

i7 : SubQ = Sub ** QR

o7 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}}
                                 2
     natural  => (QR[T   ..T   ])
                      1,1   1,2
     inclusion => | x y |

o7 : DGSubmodule

i8 : isWellDefined SubQ

o8 = true

i9 : SubQ.ambient === M ** QR

o9 = true

Ways to use this method:

DGSubmodule ** Ring -- Base change of a DG submodule along a ring map

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:1859:0.

DGSubmodule ** Ring -- Base change of a DG submodule along a ring map

Description

See also

Ways to use this method: