DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal

Operator: /
Usage:

B = A / I
Inputs:
- A, an instance of the type DGAlgebra,
- I, an instance of the type DGIdeal, A DG ideal of A. Must satisfy I.dgAlgebra === A.
Outputs:
- B, an instance of the type DGAlgebra, A DG algebra whose underlying ring is A.natural / I.natural and whose differential is the (well-defined) descent of d_A.

Description

The descent is well-defined precisely because I is d-closed: for any lift g' of a class g in B, the difference g - g' lies in I, so d_A(g) - d_A(g') \in d_A(I) \subset I and the image [d_A(g)] \in B is independent of the chosen lift.

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                    }
      Underlying algebra => R[T   ]
                               1,1
      Differential => {x}

o2 : DGAlgebra

i3 : Anat = A.natural

o3 = Anat

o3 : PolynomialRing, 1 skew commutative variable(s)

i4 : T = Anat_0

o4 = T
      1,1

o4 : Anat

i5 : I = dgIdeal(A, {T})

o5 = DGIdeal of Anat
     generators => | x T_(1,1) |

o5 : DGIdeal

i6 : B = A / I

o6 = {Ring => R                      }
                               Anat
      Underlying algebra => ---------
                            (x, T   )
                                 1,1
      Differential => {0}

o6 : DGAlgebra

i7 : isWellDefined B

o7 = true

i8 : B.natural

        Anat
o8 = ---------
     (x, T   )
          1,1

o8 : QuotientRing

Ways to use this method:

DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2811:0.

DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal

Description

See also

Ways to use this method: