DGModule / DGSubmodule -- Quotient DG module: M / S

Operator: /
Usage:

Q = M / S
Inputs:
- M, an instance of the type DGModule,
- S, an instance of the type DGSubmodule, With S.ambient === M.
Outputs:
- Q, an instance of the type DGQuotientModule,

Description

Shorthand for dgQuotientModule(M, S). See dgQuotientModule for semantics and invariants.

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 : M = freeDGModule(A, {0})

o3 = {Base ring => R               }
      DG algebra => R[T   ]
                       1,1
                                 1
      Natural module => (R[T   ])
                            1,1
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o3 : DGModule

i4 : Z = dgSubmodule(M, {})

o4 = DGSubmodule of ambient DGModule
     Degrees  => {}
     natural  => 0
     inclusion => 0

o4 : DGSubmodule

i5 : Q = M / Z

o5 = DGQuotientModule Q = M / S
                          1
     Q.natural = (R[T   ])
                     1,1
     Degrees   = {{0, 0}}

o5 : DGQuotientModule

i6 : ambient Q === M

o6 = true

i7 : isWellDefined Q

o7 = true

The zero and top-dimensional boundary cases are handled consistently: M / 0 is a DGQuotientModule wrapping a cokernel isomorphic to M, and M / M is the zero DGQuotientModule.

i8 : Sfull = dgSubmodule(M, id_(M.natural))

o8 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}}
                          1
     natural  => (R[T   ])
                     1,1
     inclusion => | 1 |

o8 : DGSubmodule

i9 : isZero (M / Sfull)

o9 = true

Ways to use this method:

DGModule / DGSubmodule -- Quotient DG module: M / S

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:3287:0.

DGModule / DGSubmodule -- Quotient DG module: M / S

Description

See also

Ways to use this method: