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

Operator: **
Usage:

fS = f ** S
Inputs:
- f, an instance of the type DGModuleMap, A DG module map M -> N over a DG algebra A.
- S, a ring,
Outputs:
- fS, an instance of the type DGModuleMap, A DG module map M ** S -> N ** S (over A ** S) whose underlying matrix is obtained entry-wise by substituting each entry of f.natural into the base-changed ring.

Description

Functorial: identityDGModuleMap(M) ** S equals identityDGModuleMap(M ** S), and composition is preserved. Cache-identity is what ensures the source and target of the result are the same cached DG modules as M ** S and N ** S.

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})

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

o3 : DGModule

i4 : idM = identityDGModuleMap M

                               1
o4 = {Source => (R[T   ..T   ]) }
                    1,1   1,2
                               1
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => | 1 |

o4 : DGModuleMap

i5 : QR = R / ideal(x^2)

o5 = QR

o5 : QuotientRing

i6 : idMQ = idM ** QR

                                1
o6 = {Source => (QR[T   ..T   ]) }
                     1,1   1,2
                                1
      Target => (QR[T   ..T   ])
                     1,1   1,2
      Natural => | 1 |

o6 : DGModuleMap

i7 : isWellDefined idMQ

o7 = true

i8 : (idMQ.source) === (M ** QR)

o8 = true

i9 : (idMQ.target) === (M ** QR)

o9 = true

Functoriality is literal at the matrix level: the natural matrix of idM ** QR is the identity matrix on (M ** QR).natural.

i10 : MQ = M ** QR

o10 = {Base ring => QR                    }
       DG algebra => QR[T   ..T   ]
                         1,1   1,2
                                         1
       Natural module => (QR[T   ..T   ])
                              1,1   1,2
       Generator degrees => {{0, 0}}
       Differentials on gens => {0}

o10 : DGModule

i11 : idMQ.natural == id_(MQ.natural)

o11 = true

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2034:0.

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

Description

See also

Ways to use this method: