DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module

Operator: **
Usage:

SubN = Sub ** N

SubN = N ** Sub
Inputs:
- Sub, an instance of the type DGSubmodule,
- N, a module, A free module over a compatible ring.
Outputs:
- SubN, an instance of the type DGSubmodule, The DG submodule of (Sub.ambient) ** N whose inclusion acts as inclusion(Sub) \otimes id_N on natural generators indexed by (i, j).

Description

Equivalently: for each natural generator of N, embed a copy of Sub's inclusion matrix into the corresponding block of the tensored ambient. The commuted form N ** Sub is defined as Sub ** 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, y_Anat}})

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

o5 : DGSubmodule

i6 : N = R^2

      2
o6 = R

o6 : R-module, free

i7 : SubN = Sub ** N

o7 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}, {0, 1}, {0, 1}}
                     4
     natural  => Anat
     inclusion => | x 0 y 0 |
                  | 0 x 0 y |

o7 : DGSubmodule

i8 : isWellDefined SubN

o8 = true

i9 : SubN.ambient === M ** N

o9 = true

Ways to use this method:

DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module
Module ** DGSubmodule

The source of this document is in DGAlgebras/doc.m2:2139:0.

DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module

Description

See also

Ways to use this method: