DGSubmodule + DGSubmodule -- Sum of two DG submodules of a common ambient

Operator: +
Usage:

U = 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 inclusion matrix is the horizontal concatenation of S.inclusion.natural and T.inclusion.natural.

Description

Preserves d-closure and contains both summands. Idempotent: S + S == S.

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 {{y_Anat}})

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

o6 : DGSubmodule

i7 : ST = S + T

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

o7 : DGSubmodule

i8 : isWellDefined ST

o8 = true

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

o9 = true

i10 : (S + S) == S

o10 = true

Caveat

If the two ambients are distinct objects (even if isomorphic), this raises an error — the operation is about lattice structure within a fixed ambient.

Ways to use this method:

DGSubmodule + DGSubmodule -- Sum of two DG submodules of a common ambient

The source of this document is in DGAlgebras/doc.m2:3790:0.

DGSubmodule + DGSubmodule -- Sum of two DG submodules of a common ambient

Description

Caveat

See also

Ways to use this method: