prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion

Function: prune
Usage:

Sp = prune S

Sp = minimalPresentation S
Inputs:
- S, an instance of the type DGSubmodule,
Optional inputs:
- Exclude => ..., default value {},
Outputs:
- Sp, an instance of the type DGSubmodule, A DG submodule of the same ambient DG module whose inclusion matrix has had A.natural-linearly redundant columns removed.

Description

prune S computes mingens image S.inclusion.natural and builds a new DG submodule from that reduced inclusion matrix. The A.natural-span of the new inclusion matrix equals the original: image Sp.inclusion.natural == image S.inclusion.natural.

A pruningMap is cached in Sp.cache as a DGModuleMap from Sp to S; when the inclusion matrix is already minimal this is the identity on S (and prune S === S).

Well-definedness (d-closure) and the underlying natural-level image are preserved by pruning.

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 {{1_Anat, x_Anat, y_Anat}})

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}, {0, 1}, {0, 1}}
                     3
     natural  => Anat
     inclusion => | 1 x y |

o5 : DGSubmodule

i6 : Sp = prune S

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

o6 : DGSubmodule

i7 : numcols (inclusion Sp).natural

o7 = 1

i8 : numcols (inclusion S).natural

o8 = 3

i9 : image (inclusion S).natural == image (inclusion Sp).natural

o9 = true

i10 : isWellDefined Sp

o10 = true

minimalPresentation is a synonym for prune on DG submodules.

Ways to use this method:

prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4624:0.

prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion

Description

See also

Ways to use this method: