DGAlgebra ** Ring -- Base change of a DG algebra to another ring

Operator: **
Usage:

B = A ** S
Inputs:
- A, an instance of the type DGAlgebra,
- S, a ring, A ring that admits substitution from A.ring. The common case is S = A.ring / I for some ideal I, but any ring into which the variables of A.natural substitute is allowed.
Outputs:
- B, an instance of the type DGAlgebra, A DG algebra over S with the same generator degrees as A and with differential obtained by substituting every entry of the differential matrix of A into B.natural.

Description

Base change: the underlying graded algebra is the same exterior / symmetric shape over S, and the differential is transported by substitute. When S = A.ring / I, this models the DG algebra "A mod I" used, for instance, in building the Koszul complex of a quotient ring.

The result is cached on A.cache: repeated calls A ** S with the same pair (A, S) return the SAME DGAlgebra object. This identity is essential for functoriality of base change on DGAlgebraMaps and DGModules — see Base change and tensor with non-DG types.

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

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA(R)

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

o4 : DGAlgebra

i5 : B === A ** S

o5 = true

i6 : Bdd = toComplex B

      1      4      6      4      1
o6 = S  <-- S  <-- S  <-- S  <-- S
                                  
     0      1      2      3      4

o6 : Complex

i7 : Bdd.dd

          1                   4
o7 = 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  |

o7 : ComplexMap

Base change along the trivial quotient R / ideal 0_R is structure- preserving: the result is a well-defined DGAlgebra over a ring isomorphic to R.

i8 : R' = ZZ/101[x]

o8 = R'

o8 : PolynomialRing

i9 : A' = koszulComplexDGA R'

o9 = {Ring => R'                    }
      Underlying algebra => R'[T   ]
                                1,1
      Differential => {x}

o9 : DGAlgebra

i10 : QR' = R' / ideal(0_(R'))

o10 = R'

o10 : PolynomialRing

i11 : isWellDefined(A' ** QR')

o11 = true

Ways to use this method:

DGAlgebra ** Ring -- Base change of a DG algebra to another ring

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:1705:0.

DGAlgebra ** Ring -- Base change of a DG algebra to another ring

Description

See also

Ways to use this method: