DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module

Operator: **
Usage:

QN = Q ** N

QN = N ** Q
Inputs:
- Q, an instance of the type DGQuotientModule,
- N, a module, A free module over a compatible ring.
Outputs:
- QN, an instance of the type DGQuotientModule, The DG quotient module (Q.ambient ** N) / (Q.subDGModule ** N), built directly inside the tensored ambient so that the ambient of the result equals Q.ambient ** N on the nose.

Description

The commuted form N ** Q is defined as Q ** N.

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 : N = R^2

      2
o7 = R

o7 : R-module, free

i8 : QN = Q ** N

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

o8 : DGQuotientModule

i9 : isWellDefined QN

o9 = true

i10 : QN.ambient === M ** N

o10 = true

Ways to use this method:

DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module
Module ** DGQuotientModule

The source of this document is in DGAlgebras/doc.m2:2177:0.

DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module

Description

See also

Ways to use this method: