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

Operator: **
Usage:

MS = M ** S
Inputs:
- M, an instance of the type DGModule, A (free) DG module over some DG algebra A.
- S, a ring, A ring admitting substitution from A.ring. Typically S = A.ring / I.
Outputs:
- MS, an instance of the type DGModule, A DG module over A ** S with the same generator multi-degrees as M and with differential obtained by substituting every entry of each differential column into the base-changed algebra.

Description

This is the base change of M along the ring map A.ring -> S, lifted to the DG level. Since A ** S and M ** S are both cached, repeated tensoring with S is idempotent up to object identity:

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

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

o3 : DGModule

i4 : QR = R / ideal(x^2)

o4 = QR

o4 : QuotientRing

i5 : MS = M ** QR

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

o5 : DGModule

i6 : MS === M ** QR

o6 = true

i7 : MS.dgAlgebra === A ** QR

o7 = true

i8 : isWellDefined MS

o8 = true

When M is the Koszul complex on a sequence and the map A.ring -> S is a quotient map, M ** S is the Koszul complex of the same sequence computed over S.

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:1821:0.

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

Description

See also

Ways to use this method: