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annihilator(DGQuotientModule) -- The DG ideal of A annihilating a DG quotient module

Description

Computes annihilator Q.natural and wraps it as a DG ideal. As with annihilator(DGSubmodule) the result is automatically d-closed: the argument showing closure uses Leibniz together with the fact that d on the ambient descends to d on the quotient.

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 : Q = M / S

o6 = DGQuotientModule Q = M / S
     Q.natural = cokernel | x |
     Degrees   = {{0, 0}}

o6 : DGQuotientModule
 
i7 : I = annihilator Q

o7 = DGIdeal of Anat
     generators => | x |

o7 : DGIdeal
 
i8 : isWellDefined I

o8 = true
 
i9 : isSubset(dgIdeal(A, {x_Anat}), I)

o9 = true

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:5323:0.