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

Operator: **
Usage:

P = M ** N

P = N ** M
Inputs:
- M, an instance of the type DGModule, Must be semifree: M.natural must be a free A.natural-module.
- N, a module, An ordinary module over A.ring (or a compatible ring) that is also free.
Outputs:
- P, an instance of the type DGModule, A DG module over the same DG algebra A, with rank P.natural = (rank M.natural) * (rank N) generators. The multi-degree of the (i, j)-th natural generator is M.Degrees#i + (0, (degrees N)#j) (homological degree from M, internal degrees shifted by the internal degree of the j-th generator of N). The differential is d_M \otimes id_N.

Description

This is the exterior tensor product of DG-modules, where N is viewed as a complex concentrated in homological degree 0 with zero differential. The commuted form N ** M is defined as M ** 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 : 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 : N = R^2

      2
o4 = R

o4 : R-module, free

i5 : MN = M ** N

o5 = {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}, {0, 0}}
      Differentials on gens => {0, 0}

o5 : DGModule

i6 : class MN

o6 = DGModule

o6 : Type

i7 : numgens MN

o7 = 2

i8 : isWellDefined MN

o8 = true

The special case M ** R^1 is canonically isomorphic to M, and M ** R^{d} is a shift of M by the internal degree d.

Multi-degree M tensors predictably: if M has multiple homological degrees and N is free of rank r, the result has (rank M.natural) * r natural generators and is well-defined.

i9 : M2m = freeDGModule(A, {0, 1})

o9 = {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}

o9 : DGModule

i10 : N3 = R^3

       3
o10 = R

o10 : R-module, free

i11 : M2N = M2m ** N3

o11 = {Base ring => R                                                       }
       DG algebra => R[T   ..T   ]
                        1,1   1,2
                                        6
       Natural module => (R[T   ..T   ])
                             1,1   1,2
       Generator degrees => {{0, 0}, {0, 0}, {0, 0}, {1, 0}, {1, 0}, {1, 0}}
       Differentials on gens => {0, 0, 0, 0, 0, 0}

o11 : DGModule

i12 : numgens M2N == 2 * 3

o12 = true

i13 : isWellDefined M2N

o13 = true

Edge case: tensoring with the zero module R^0 gives a well-defined zero DGModule.

i14 : MZ = M ** R^0

o14 = {Base ring => R             }
       DG algebra => R[T   ..T   ]
                        1,1   1,2
       Natural module => 0
       Generator degrees => {}
       Differentials on gens => {}

o14 : DGModule

i15 : isWellDefined MZ

o15 = true

i16 : numgens MZ == 0

o16 = true

Caveat

Both M and N must be free (semifree on the DG side). For the non-free case, take an initial free resolution and then tensor.

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2100:0.

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

Description

Caveat

See also

Ways to use this method: