DGQuotientModule -- The class of quotient DG modules M / S

Description

A DGQuotientModule Q = M / S is the cokernel of the inclusion S.inclusion : S → M, equipped with the induced differential. Unlike DGSubmodule, DGQuotientModule is NOT a subtype of DGModule: its .natural is a cokernel module, which does not support the free-module operations (rank, basis by index) that DG algorithms assume on a generic DGModule.

Beyond the DGModule-like keys (.dgAlgebra, .natural, .Degrees, .diff), a DGQuotientModule carries:

Q.ambient — the DGModule M from which Q was built; accessible via ambient.

Q.subDGModule — the killed submodule S; accessible via subDGModule.

Q.projection — the quotient map M → Q as a DGModuleMap; accessible via projection.

New DGQuotientModules are built by dgQuotientModule or the shorthand M / S.

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 : Z = dgSubmodule(M, {})

o4 = DGSubmodule of ambient DGModule
     Degrees  => {}
     natural  => 0
     inclusion => 0

o4 : DGSubmodule

i5 : Q = M / Z

o5 = DGQuotientModule Q = M / S
                          1
     Q.natural = (R[T   ])
                     1,1
     Degrees   = {{0, 0}}

o5 : DGQuotientModule

i6 : instance(Q, DGQuotientModule)

o6 = true

i7 : ambient Q === M

o7 = true

i8 : subDGModule Q === Z

o8 = true

i9 : source projection Q === M

o9 = true

i10 : target projection Q === Q

o10 = true

Functions and methods returning an object of class DGQuotientModule:

dgQuotientModule -- Construct the quotient DG module M / S

Methods that use an object of class DGQuotientModule:

ambient(DGQuotientModule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
projection(DGQuotientModule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
subDGModule(DGQuotientModule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
annihilator(DGQuotientModule) -- The DG ideal of A annihilating a DG quotient module
degrees(DGQuotientModule) -- see degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module
Module ** DGQuotientModule -- see DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module
DGQuotientModule ** Ring -- Base change of a DG quotient module along a ring map
diff(DGQuotientModule,Vector) -- Apply the induced differential on a DG quotient module to an element
HH DGQuotientModule -- The graded homology of a DG quotient module
homologyModule(DGQuotientModule) -- see HH DGQuotientModule -- The graded homology of a DG quotient module
HH_ZZ DGQuotientModule -- The degree-n homology of a DG quotient module
isAcyclic(DGQuotientModule) -- Determine whether a DG quotient module has vanishing positive-degree homology
isHomogeneous(DGQuotientModule) -- see isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
isWellDefined(DGQuotientModule) -- Check that a DG quotient module has a well-defined differential
isZero(DGQuotientModule) -- see isZero(DGModule) -- Does the DG object have no natural generators?
maxDegree(DGQuotientModule) -- Largest hom-degree present in a DG quotient module
net(DGQuotientModule) (missing documentation)
numgens(DGQuotientModule) -- see numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
minimalPresentation(DGQuotientModule) -- see prune(DGQuotientModule) -- Minimize the presentation of a DG quotient module
prune(DGQuotientModule) -- Minimize the presentation of a DG quotient module
cover(DGQuotientModule) -- see super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
relations(DGQuotientModule) -- see super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
super(DGQuotientModule) -- see super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
toComplex(DGQuotientModule) -- Export a DG quotient module to a Complex

For the programmer

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:3024:0.

DGQuotientModule -- The class of quotient DG modules M / S

Description

See also

Functions and methods returning an object of class DGQuotientModule:

Methods that use an object of class DGQuotientModule:

For the programmer