annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule

Function: annihilator (missing documentation)
Usage:

I = annihilator S
Inputs:
- S, an instance of the type DGSubmodule,
Optional inputs:
- Strategy (missing documentation) => ..., default value null,
Outputs:
- I, an instance of the type DGIdeal, The DG ideal of S.ambient.dgAlgebra whose underlying ideal is the annihilator of image S.inclusion.natural as a submodule of S.ambient.natural.

Description

Computes annihilator(image S.inclusion.natural) at the A.natural level, then wraps the result as a DG ideal. The result is a DG ideal: if a annihilates S and s \in S, then d(a s) = d(a) s \pm a d(s) vanishes because both terms vanish (the second since d(s) \in S), so d(a) s = 0 for every s \in S and d(a) is again in the annihilator.

i1 : R = ZZ/101[x, y]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o2 : DGAlgebra

i3 : Anat = A.natural

o3 = Anat

o3 : PolynomialRing, 2 skew commutative variable(s)

i4 : M = freeDGModule(A, {0})

o4 = {Base ring => R               }
      DG algebra => Anat
                            1
      Natural module => Anat
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o4 : DGModule

i5 : S = dgSubmodule(M, matrix {{x_Anat}})

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}}
                     1
     natural  => Anat
     inclusion => | x |

o5 : DGSubmodule

i6 : I = annihilator S

o6 = DGIdeal of Anat
     generators => 0

o6 : DGIdeal

i7 : isWellDefined I

o7 = true

The annihilator of the zero submodule is the unit ideal.

i8 : S0 = dgSubmodule(M, map(M.natural, (Anat)^0, 0))

o8 = DGSubmodule of ambient DGModule
     Degrees  => {}
     natural  => 0
     inclusion => 0

o8 : DGSubmodule

i9 : annihilator S0 == dgIdeal(A, {1_Anat})

o9 = true

Ways to use this method:

annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:5284:0.

annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule

Description

See also

Ways to use this method: