DGAlgebraMap ** Ring -- Base change of a DG algebra map along a ring map

Operator: **
Usage:

phiS = phi ** S
Inputs:
- phi, an instance of the type DGAlgebraMap, A DG algebra map A -> B.
- S, a ring,
Outputs:
- phiS, an instance of the type DGAlgebraMap, A DG algebra map A ** S -> B ** S whose underlying RingMap sends each generator of A.natural to the image under phi, substituted into (B ** S).natural.

Description

Functorial and compatible with object identity: because A ** S and B ** S are cached, repeated evaluations of phi ** S land in the same source/target, and identityDGAlgebraMap(A) ** S equals identityDGAlgebraMap(A ** 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 : idA = identityDGAlgebraMap A

o3 = map (R[T   ..T   ], R[T   ..T   ], {T   , T   , x, y})
             1,1   1,2      1,1   1,2     1,1   1,2

o3 : DGAlgebraMap

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

o4 = QR

o4 : QuotientRing

i5 : idAQ = idA ** QR

o5 = map (QR[T   ..T   ], QR[T   ..T   ], {T   , T   , x, y})
              1,1   1,2       1,1   1,2     1,1   1,2

o5 : DGAlgebraMap

i6 : isWellDefined idAQ

o6 = true

i7 : (idAQ.source) === (A ** QR)

o7 = true

i8 : (idAQ.target) === (A ** QR)

o8 = true

Ways to use this method:

DGAlgebraMap ** Ring -- Base change of a DG algebra 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.m2:1990:0.

DGAlgebraMap ** Ring -- Base change of a DG algebra map along a ring map

Description

See also

Ways to use this method: