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ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule

Description

These accessors navigate between the DG types in the DGModule / DGSubmodule / DGQuotientModule hierarchy:

ambient on a DGIdeal returns the enclosing DGAlgebra; on a DGSubmodule or DGQuotientModule it returns the enclosing DGModule.

inclusion on a DGSubmodule S returns the DGModuleMap S → ambient(S).

projection on a DGQuotientModule Q returns the DGModuleMap ambient(Q) → Q.

subDGModule on a DGQuotientModule Q returns the DGSubmodule that was killed to form Q.

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 : S = dgSubmodule(M, id_(M.natural))

o4 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}}
                          1
     natural  => (R[T   ])
                     1,1
     inclusion => | 1 |

o4 : DGSubmodule
 
i5 : Q = M / S

o5 = DGQuotientModule Q = M / S
     Q.natural = cokernel | 1 |
     Degrees   = {{0, 0}}

o5 : DGQuotientModule
 
i6 : ambient S === M

o6 = true
 
i7 : ambient Q === M

o7 = true
 
i8 : source inclusion S === S

o8 = true
 
i9 : target inclusion S === M

o9 = true
 
i10 : source projection Q === M

o10 = true
 
i11 : target projection Q === Q

o11 = true
 
i12 : subDGModule Q === S

o12 = true

For a DGIdeal the analogous accessor returns the DGAlgebra:

i13 : I = dgIdeal(A, {x_(A.natural)})

o13 = DGIdeal of R[T   ]
                    1,1
      generators => | x |

o13 : DGIdeal
 
i14 : ambient I === A

o14 = true

See also

Ways to use ambient:

For the programmer

The object ambient is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:3343:0.