DGModule ** DGModule -- Exterior tensor product of DG modules over DG algebras sharing a ground ring

Operator: **
Usage:

P = M ** N
Inputs:
- M, an instance of the type DGModule, A (free) DG module over some DG algebra A.
- N, an instance of the type DGModule, A (free) DG module over some DG algebra B, with A.ring === B.ring.
Outputs:
- P, an instance of the type DGModule, A (free) DG module over A ** B whose natural generators are indexed lexicographically by (i, j) with i ranging over M-generators and j over N-generators. The multi-degree of (i, j) is M.Degrees#i + N.Degrees#j coordinate-wise, and the differential is $d(e_i \otimes f_j) = d_M(e_i) \otimes f_j + (-1)^{|e_i|}\, e_i \otimes d_N(f_j)$.

Description

This is the EXTERIOR tensor product: we do not quotient by an A- or B-action. The resulting DGModule lives over A ** B via the canonical inclusions. The result is cached on M.cache, so M ** N returns the same object on repeat.

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, 1})

o3 = {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}

o3 : DGModule

i4 : ng = apply(rank M.natural, i -> (M.natural)_i)

o4 = {| 1 |, | 0 |}
      | 0 |  | 1 |

o4 : List

i5 : setDiff(M, {0, x * ng#0})

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}, {1, 0}}
      Differentials on gens => {0, | x |}
                                   | 0 |

o5 : DGModule

i6 : P = M ** M

o6 = {Base ring => R                                       }
      DG algebra => R[T   ..T   ]
                       1,1   1,4
                                       4
      Natural module => (R[T   ..T   ])
                            1,1   1,4
      Generator degrees => {{0, 0}, {1, 0}, {1, 0}, {2, 0}}
      Differentials on gens => {0, | x |, | x |, |  0 |}
                                   | 0 |  | 0 |  |  x |
                                   | 0 |  | 0 |  | -x |
                                   | 0 |  | 0 |  |  0 |

o6 : DGModule

i7 : P.dgAlgebra === (A ** A)

o7 = true

Ways to use this method:

DGModule ** DGModule -- Exterior tensor product of DG modules over DG algebras sharing a ground ring

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:706:0.

DGModule ** DGModule -- Exterior tensor product of DG modules over DG algebras sharing a ground ring

Description

See also

Ways to use this method: