DGModule -- The class of DG modules over a DG algebra

Description

A DGModule M over a DGAlgebra A is a free graded A.natural-module together with a differential that squares to zero and satisfies the Leibniz rule against A. Internally M is a hashtable with keys:

M.dgAlgebra — the ambient DGAlgebra.

M.natural — the underlying graded A.natural-module.

M.Degrees — the list of multi-degrees of the natural generators.

M.diff — a list of differentials, one per natural generator, each living in M.natural in the appropriate homological degree.

New DGModules are constructed by freeDGModule; the differential on generators is set via setDiff. Every DGSubmodule is also a DGModule (as a subtype); by contrast a DGQuotientModule is a separate type with a compatible presentation.

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, 1})

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

o3 : DGModule

i4 : natGens = apply(rank M.natural, i -> (M.natural)_i)

o4 = {| 1 |, | 0 |}
      | 0 |  | 1 |

o4 : List

i5 : setDiff(M, {0, x * natGens#0})

o5 = {Base ring => R                       }
      DG algebra => R[T   ]
                       1,1
                                 2
      Natural module => (R[T   ])
                            1,1
      Generator degrees => {{0, 0}, {1, 0}}
      Differentials on gens => {0, | x |}
                                   | 0 |

o5 : DGModule

i6 : isWellDefined M

o6 = true

i7 : M.Degrees

o7 = {{0, 0}, {1, 0}}

o7 : List

Types of DGModule:

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

Functions and methods returning an object of class DGModule:

adjoinGenerators -- Adjoin new free generators to a DG module with prescribed differentials
freeDGModule -- Construct a free DG module over a DG algebra
killCycles(DGModule) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module
koszulComplexDGM -- The Koszul complex of a module as a DG module
minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
semifreeResolution -- Build a semifree DG module resolving a module over the base ring
setDiff(DGModule,List) -- Install the list of generator differentials on a free DG module

Methods that use an object of class DGModule:

adjoinGenerators(DGModule,List) -- see adjoinGenerators -- Adjoin new free generators to a DG module with prescribed differentials
components(DGModule) (missing documentation)
degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
dgComplex(DGModule) -- see dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
DGIdeal * DGModule (missing documentation)
DGModule ** DGModule -- Exterior tensor product of DG modules over DG algebras sharing a ground ring
DGModule ** Module -- Exterior tensor product of a DG module with an ordinary free module
Module ** DGModule -- see DGModule ** Module -- Exterior tensor product of a DG module with an ordinary free module
DGModule ** Ring -- Base change of a DG module along a ring map
DGModule ++ DGModule (missing documentation)
DGModule / DGSubmodule -- Quotient DG module: M / S
dgModuleMap(DGModule,DGModule,List) -- see dgModuleMap -- Construct a DGModuleMap from a matrix or from a list of image Vectors
dgModuleMap(DGModule,DGModule,Matrix) -- see dgModuleMap -- Construct a DGModuleMap from a matrix or from a list of image Vectors
map(DGModule,DGModule,ZZ) -- see DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
dgModuleSummary(DGModule) -- see dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
dgModuleSummary(DGModule,ZZ) -- see dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
dgQuotientModule(DGModule,DGSubmodule) -- see dgQuotientModule -- Construct the quotient DG module M / S
dgSubmodule(DGModule,List) -- see dgSubmodule -- Construct a d-closed submodule of a DG module
dgSubmodule(DGModule,Matrix) -- see dgSubmodule -- Construct a d-closed submodule of a DG module
diff(DGModule,Vector) -- Apply the DG module differential to an element
differential(DGModule) -- see differential -- The list of generator differentials stored in a DG algebra or DG module
directSum(DGModule) (missing documentation)
displayModuleBlockDiff(DGModule,List,List) -- see displayModuleBlockDiff -- Pretty-print the labeled block matrix of a DG module differential
displayModuleBlockDiff(DGModule,ZZ) -- see displayModuleBlockDiff -- Pretty-print the labeled block matrix of a DG module differential
ensureDGAlgebraCaches(DGModule) -- see ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
generatorDegrees(DGModule) -- see generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
generatorTable(DGModule) -- see generatorTable -- Display the generator list of a DG module with hom-degrees and differentials
getBasis(ZZ,DGModule) -- Basis of the hom-degree-n piece of a DG module
getBoundaryPreimage(DGModule,List) -- Lift a list of boundaries sharing a hom-degree through the module differential
getBoundaryPreimage(DGModule,Vector) -- Lift a boundary in a DG module to a preimage under the differential
HH DGModule -- The graded homology of a DG module as a module over HH(A)
homologyModule(DGModule) -- see HH DGModule -- The graded homology of a DG module as a module over HH(A)
HH_ZZ DGModule -- The degree-n homology of a DG module as a module over the base ring
homologyClass(DGModule,Vector) -- The homology class of a cycle in a DG module
identityDGModuleMap(DGModule) -- see identityDGModuleMap -- The identity DGModuleMap on a DG module
invalidateDGAlgebraCache(DGModule) -- see invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
isAcyclic(DGModule) -- Determine whether a DG module has vanishing positive-degree homology
isDGSubmodule(DGModule,Matrix) -- see isDGSubmodule -- Test whether the image of an inclusion matrix is d-closed
isFreeDGModule(DGModule) -- see isFreeDGModule -- Is the underlying A.natural-module free?
isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
isMinimalSemifreeResolution(DGModule) -- see isMinimalSemifreeResolution -- Test whether a semifree DG module is minimal over its DG algebra
isValidDGModule(DGModule) -- see isValidDGModule -- Structural-invariant check on a DG module
isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero
isWellDefinedDifferential(DGModule) -- see isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module
isZero(DGModule) -- Does the DG object have no natural generators?
liftToDGModuleMap(DGModule,DGModule,List) -- see liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap
liftToDGModuleMap(DGModule,DGModule,Matrix) -- see liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap
liftToDGModuleMap(DGModule,DGModule,Vector) -- see liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap
maxDegree(DGModule) -- Largest hom-degree present in a DG module
moduleBlockDiff(DGModule,ZZ) -- see moduleBlockDiff -- The hom-degree-n differential of a DG module as a labeled block matrix
moduleDifferential(ZZ,DGModule) -- see moduleDifferential -- The hom-degree-n differential of a DG module as a matrix over the base ring
net(DGModule) (missing documentation)
numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
minimalPresentation(DGModule) -- see prune(DGModule) -- Pruning a DG module is the identity
prune(DGModule) -- Pruning a DG module is the identity
rank(DGModule) -- Rank of the underlying free A.natural-module
ring(DGModule) (missing documentation)
toComplex(DGModule) -- Export a DG module to a Complex of free base-ring modules
toComplex(DGModule,ZZ) -- Export a DG module to a Complex with an explicit hom-degree bound
underlyingAlgebra(DGModule) -- see underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
underlyingRing(DGModule) -- see underlyingRing -- The commutative base ring of a DG algebra or DG module
zeroDGModuleMap(DGModule,DGModule) -- see zeroDGModuleMap -- The zero DGModuleMap between two DG modules over a common algebra

For the programmer

The object DGModule 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.m2:2935:0.