DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring

Operator: **
Usage:

C = A ** B

(A1, A2) = tensorFactors C

(iotaA, iotaB) = tensorInclusions C
Inputs:
- A, an instance of the type DGAlgebra,
- B, an instance of the type DGAlgebra, Must satisfy A.ring === B.ring.
Outputs:
- C, an instance of the type DGAlgebra, A DGAlgebra over A.ring whose underlying algebra has #A.Degrees + #B.Degrees generators, with the obvious multi-degrees, and whose differential is d_A on the A-generators and d_B on the B-generators (transported via the canonical inclusions).

Description

This is the exterior tensor product of A and B as DGAs: both are viewed as augmented DG algebras over their common ground ring, and the tensor is taken componentwise in multi-degrees. Internally the result is cached on A.cache, so C1 = A ** B and C2 = A ** B return the SAME DGAlgebra object, which is essential for composition identities to hold downstream.

The helpers tensorFactors and tensorInclusions recover the pair (A, B) and the canonical inclusions A -> C and B -> C (as DGAlgebraMaps).

i1 : R = QQ[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 : C = A ** A

o3 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,4
      Differential => {x, y, x, y}

o3 : DGAlgebra

i4 : numgens C.natural

o4 = 4

i5 : (iotaA, iotaB) = tensorInclusions C

o5 = (map (R[T   ..T   ], R[T   ..T   ], {T   , T   , x, y}), map
              1,1   1,4      1,1   1,2     1,1   1,2             
     ------------------------------------------------------------------------
     (R[T   ..T   ], R[T   ..T   ], {T   , T   , x, y}))
         1,1   1,4      1,1   1,2     1,3   1,4

o5 : Sequence

Ways to use this method:

DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:643:0.

DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring

Description

See also

Ways to use this method: