intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient

Function: intersect
Usage:

U = intersect(S, T)
Inputs:
- S, an instance of the type DGSubmodule,
- T, an instance of the type DGSubmodule, With S.ambient === T.ambient.
Outputs:
- U, an instance of the type DGSubmodule, The DGSubmodule of S.ambient whose image is image(S.inclusion.natural) \cap image(T.inclusion.natural).

Description

Preserves d-closure directly. Intersecting with the zero submodule yields the zero submodule; intersecting with the same submodule is idempotent (intersect(S, S) == S as submodules).

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 : M = freeDGModule(A, {0})

o3 = {Base ring => R                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       1
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o3 : DGModule

i4 : Anat = A.natural

o4 = Anat

o4 : PolynomialRing, 2 skew commutative variable(s)

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

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

o5 : DGSubmodule

i6 : T = dgSubmodule(M, matrix {{x_Anat * y_Anat}})

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 2}}
                     1
     natural  => Anat
     inclusion => | xy |

o6 : DGSubmodule

i7 : cap = intersect(S, T)

o7 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 2}}
                     1
     natural  => Anat
     inclusion => | xy |

o7 : DGSubmodule

i8 : isWellDefined cap

o8 = true

i9 : isSubset(cap, S) and isSubset(cap, T)

o9 = true

Boundary case: intersection with the zero submodule is zero.

i10 : Z = dgSubmodule(M, map(M.natural, (Anat)^0, 0))

o10 = DGSubmodule of ambient DGModule
      Degrees  => {}
      natural  => 0
      inclusion => 0

o10 : DGSubmodule

i11 : isZero intersect(S, Z)

o11 = true

Ways to use this method:

intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient

The source of this document is in DGAlgebras/doc.m2:3830:0.

intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient

Description

See also

Ways to use this method: