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DGSubmodule -- The class of d-closed submodules of a DG module

Description

A DGSubmodule S of a DGModule M is a free A.natural-module together with an inclusion S.inclusion : S → M of DGModules whose column span in M.natural is closed under the differential of M. DGSubmodule is a subtype of DGModule, so every DG-module operation applies to submodules as well.

Beyond the DGModule keys, a DGSubmodule carries:

S.ambient — the enclosing DGModule M.

S.inclusion — a DGModuleMap S → M; use inclusion to access it.

New DGSubmodules are constructed by dgSubmodule, which takes either an inclusion matrix or a list of generators, and d-saturates the input automatically. See Operations on DG Submodules for the lattice operations (sum, intersection, equality, containment).

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing
 
i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra
 
i3 : M = freeDGModule(A, {0})

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

o3 : DGModule
 
i4 : S = dgSubmodule(M, id_(M.natural))

o4 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}}
                          1
     natural  => (R[T   ])
                     1,1
     inclusion => | 1 |

o4 : DGSubmodule
 
i5 : ambient S === M

o5 = true
 
i6 : source inclusion S === S

o6 = true
 
i7 : target inclusion S === M

o7 = true

See also

Functions and methods returning an object of class DGSubmodule:

Methods that use an object of class DGSubmodule:

  • ambient(DGSubmodule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
  • inclusion(DGSubmodule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
  • annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule
  • degrees(DGSubmodule) -- see degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
  • DGIdeal * DGSubmodule (missing documentation)
  • DGModule / DGSubmodule -- Quotient DG module: M / S
  • dgQuotientModule(DGModule,DGSubmodule) -- see dgQuotientModule -- Construct the quotient DG module M / S
  • DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module
  • Module ** DGSubmodule -- see DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module
  • DGSubmodule ** Ring -- Base change of a DG submodule along a ring map
  • DGSubmodule + DGSubmodule -- Sum of two DG submodules of a common ambient
  • DGSubmodule == DGSubmodule -- Equality and containment of DG submodules
  • isSubset(DGSubmodule,DGSubmodule) -- see DGSubmodule == DGSubmodule -- Equality and containment of DG submodules
  • isFreeDGModule(DGSubmodule) -- see isFreeDGModule -- Is the underlying A.natural-module free?
  • isHomogeneous(DGSubmodule) -- see isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
  • isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module
  • isZero(DGSubmodule) -- see isZero(DGModule) -- Does the DG object have no natural generators?
  • module(DGSubmodule) (missing documentation)
  • net(DGSubmodule) (missing documentation)
  • numgens(DGSubmodule) -- see numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
  • minimalPresentation(DGSubmodule) -- see prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion
  • prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion
  • rank(DGSubmodule) -- see rank(DGModule) -- Rank of the underlying free A.natural-module
  • super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module

For the programmer

The object DGSubmodule is a type, with ancestor classes DGModule < MutableHashTable < HashTable < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2977:0.