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DGAlgebras :: DGAlgebra ** Ring

DGAlgebra ** Ring -- Tensor product of a DGAlgebra and another ring.

Synopsis

Description

Tensor product of a DGAlgebra and another ring (typically a quotient of A.ring).

i1 : R = ZZ/101[a,b,c,d]

o1 = R

o1 : PolynomialRing
i2 : A = koszulComplexDGA(R)

o2 = {Ring => R                      }
      Underlying algebra => R[T ..T ]
                               1   4
      Differential => {a, b, c, d}

o2 : DGAlgebra
i3 : S = R/ideal{a^3,a*b*c}

o3 = S

o3 : QuotientRing
i4 : B = A ** S

o4 = {Ring => S                      }
      Underlying algebra => S[T ..T ]
                               1   4
      Differential => {a, b, c, d}

o4 : DGAlgebra
i5 : Bdd = toComplex B

      1      4      6      4      1
o5 = S  <-- S  <-- S  <-- S  <-- S
                                  
     0      1      2      3      4

o5 : ChainComplex
i6 : Bdd.dd

          1                   4
o6 = 0 : S  <--------------- S  : 1
               | a b c d |

          4                                 6
     1 : S  <----------------------------- S  : 2
               {1} | -b -c 0  -d 0  0  |
               {1} | a  0  -c 0  -d 0  |
               {1} | 0  a  b  0  0  -d |
               {1} | 0  0  0  a  b  c  |

          6                           4
     2 : S  <----------------------- S  : 3
               {2} | c  d  0  0  |
               {2} | -b 0  d  0  |
               {2} | a  0  0  d  |
               {2} | 0  -b -c 0  |
               {2} | 0  a  0  -c |
               {2} | 0  0  a  b  |

          4                  1
     3 : S  <-------------- S  : 4
               {3} | -d |
               {3} | c  |
               {3} | -b |
               {3} | a  |

o6 : ChainComplexMap