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DGAlgebras -- Data types and basic functions on differential graded (DG) Algebras.

Description

This package is used to define and manipulate DG algebras.

Contributors

Some documentation was added by Daniel Rostamloo and David Eisenbud.

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Authors

Version

This documentation describes version 2.0 of DGAlgebras, released April 27, 2026.

Citation

If you have used this package in your research, please cite it as follows:

@misc{DGAlgebrasSource,
  title = {{DGAlgebras: Data type for DG algebras. Version~2.0}},
  author = {Frank Moore and Keller VandeBogert},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

  • Types
    • DGAlgebra -- The class of all DGAlgebras
    • DGAlgebraMap -- The class of DG algebra maps
    • DGIdeal -- The class of d-closed graded ideals of a DG algebra
    • DGModule -- The class of DG modules over a DG algebra
    • DGModuleMap -- The class of morphisms of DG modules over a common DG algebra
    • DGQuotientModule -- The class of quotient DG modules M / S
    • DGSubmodule -- The class of d-closed submodules of a DG module
  • Functions and commands
    • acyclicClosure -- Tate's construction of a semifree acyclic DG algebra extension
    • adjoinGenerators -- Adjoin new free generators to a DG module with prescribed differentials
    • adjoinVariables -- Extend a DG algebra by a user-chosen list of cycles to be killed
    • inclusion -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
    • projection -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
    • subDGModule -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
    • baseChange -- Transport a DG algebra along a ring map on its base
    • blockDiff -- prepares a map for display
    • deviations -- Computes the deviations of the input ring, complex, or power series.
    • deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
    • tensorFactors -- see DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
    • tensorInclusions -- see DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
    • dgAlgebraMap -- Define a DG algebra map between DG algebras.
    • dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
    • dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
    • dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
    • dgModuleMap -- Construct a DGModuleMap from a matrix or from a list of image Vectors
    • dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
    • dgQuotientModule -- Construct the quotient DG module M / S
    • dgSubmodule -- Construct a d-closed submodule of a DG module
    • differential -- The list of generator differentials stored in a DG algebra or DG module
    • displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • displayModuleBlockDiff -- Pretty-print the labeled block matrix of a DG module differential
    • ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
    • expandGeomSeries -- Expand a geometric series to a specified degree.
    • findNaryTrivialMasseyOperation -- see findTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)
    • findTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)
    • freeDGAlgebra -- Construct a free DG algebra with given generator degrees
    • freeDGModule -- Construct a free DG module over a DG algebra
    • generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
    • generatorTable -- Display the generator list of a DG module with hom-degrees and differentials
    • getBasis -- Get a basis for a particular homological degree of a DG algebra.
    • getBoundaryPreimage -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
    • getDegNModule -- Extract the degree-n strand of a graded algebra as a module over its degree-zero piece
    • getGenerators -- Cycles whose classes generate the homology algebra of a DG algebra
    • homologyAlgebra -- Compute the homology algebra of a DGAlgebra.
    • homologyClass -- Computes the element of the homology algebra corresponding to a cycle in a DGAlgebra.
    • homologyModule -- Compute the homology of a DGModule as a module over a DGAlgebra.
    • identityDGAlgebraMap -- The identity DGAlgebraMap on a DG algebra
    • identityDGModuleMap -- The identity DGModuleMap on a DG module
    • invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
    • isAcyclic -- Determine whether a DG algebra has no higher homology
    • isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A
    • isDGSubmodule -- Test whether the image of an inclusion matrix is d-closed
    • isFreeDGModule -- Is the underlying A.natural-module free?
    • isGolod -- Determines if a ring is Golod
    • isGolodHomomorphism -- Determines if the canonical map from the ambient ring is Golod
    • isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial
    • isMinimalSemifreeResolution -- Test whether a semifree DG module is minimal over its DG algebra
    • isValidDGAlgebra -- Structural invariant check for a DG algebra
    • isValidDGModule -- Structural-invariant check on a DG module
    • isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module
    • isZero -- see isZero(DGModule) -- Does the DG object have no natural generators?
    • killCycles -- Adjoin variables to kill non-bounding cycles in the lowest positive degree
    • killHomologyAtDegree -- Adjoin variables killing the homology of a DG algebra at a prescribed degree
    • koszulComplexDGA -- The Koszul complex on the variables of a ring, as a DG algebra
    • koszulComplexDGM -- The Koszul complex of a module as a DG module
    • liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
    • liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap
    • masseyTripleProduct -- Computes the Massey triple product of a set of cycles or homology classes
    • maxDegree -- Computes the maximum homological degree of a DGAlgebra
    • minimalModel -- Build the minimal DG algebra resolution of a quotient ring
    • minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
    • moduleBlockDiff -- The hom-degree-n differential of a DG module as a labeled block matrix
    • moduleDifferential -- The hom-degree-n differential of a DG module as a matrix over the base ring
    • polyDifferential -- Matrix and element-level differentials of a DG algebra
    • polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra
    • restrictDifferential -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
    • semifreeResolution -- Build a semifree DG module resolving a module over the base ring
    • setDiff -- Install or replace the differential on a DG algebra
    • subDGAlgebra -- The sub-DG algebra generated by a chosen subset of DG generators
    • toComplex -- Converts a DGAlgebra to a Complex
    • toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap
    • torAlgebra -- Computes the Tor algebra of a ring
    • torMap -- Compute the map of Tor algebras associated to a RingMap.
    • truncateGenerators -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
    • underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
    • underlyingRing -- The commutative base ring of a DG algebra or DG module
    • zeroDGModuleMap -- The zero DGModuleMap between two DG modules over a common algebra
    • zerothHomology -- The zeroth homology of a DG algebra as a ring
  • Methods
    • - DGModuleMap (missing documentation)
    • acyclicClosure(DGAlgebra) -- see acyclicClosure -- Tate's construction of a semifree acyclic DG algebra extension
    • acyclicClosure(Ring) -- Tate's acyclic closure of the residue field, starting from the Koszul complex
    • adjoinGenerators(DGModule,List) -- see adjoinGenerators -- Adjoin new free generators to a DG module with prescribed differentials
    • adjoinVariables(DGAlgebra,List) -- see adjoinVariables -- Extend a DG algebra by a user-chosen list of cycles to be killed
    • ambient(DGIdeal) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
    • ambient(DGQuotientModule) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
    • 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
    • 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
    • annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule
    • baseChange(DGAlgebra,Ring) -- see baseChange -- Transport a DG algebra along a ring map on its base
    • baseChange(DGAlgebra,RingMap) -- see baseChange -- Transport a DG algebra along a ring map on its base
    • blockDiff(DGAlgebra,ZZ) -- see blockDiff -- prepares a map for display
    • cokernel(DGModuleMap) -- Cokernel of a DG module map as a DG quotient module of its target
    • components(DGModule) (missing documentation)
    • degree(DGModuleMap) (missing documentation)
    • degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
    • degrees(DGQuotientModule) -- see degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
    • degrees(DGSubmodule) -- see degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
    • deviations(Complex) -- see deviations -- Computes the deviations of the input ring, complex, or power series.
    • deviations(Ring) -- see deviations -- Computes the deviations of the input ring, complex, or power series.
    • deviations(RingElement,List) -- see deviations -- Computes the deviations of the input ring, complex, or power series.
    • deviationsToPoincare(HashTable) -- see deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
    • DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
    • tensorFactors(DGAlgebra) -- see DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
    • tensorInclusions(DGAlgebra) -- see DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
    • DGAlgebra ** Ring -- Base change of a DG algebra to another ring
    • DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal
    • dgAlgebraMap(DGAlgebra,DGAlgebra,Matrix) -- see dgAlgebraMap -- Define a DG algebra map between DG algebras.
    • isWellDefined(DGAlgebraMap) -- see dgAlgebraMap -- Define a DG algebra map between DG algebras.
    • DGAlgebraMap * DGAlgebraMap -- Composition of DG algebra maps
    • DGAlgebraMap ** DGAlgebraMap -- Tensor product of DG algebra maps
    • DGAlgebraMap ** Ring -- Base change of a DG algebra map along a ring map
    • DGAlgebraMap == DGAlgebraMap (missing documentation)
    • dgAlgebraMultMap(DGAlgebra,RingElement) -- see dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
    • dgComplex(DGAlgebra) -- see dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
    • dgComplex(DGModule) -- see dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
    • dgIdeal(DGAlgebra,Ideal) -- see dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
    • dgIdeal(DGAlgebra,List) -- see dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
    • dgIdeal(DGAlgebra,Matrix) -- see dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
    • DGIdeal * DGIdeal -- Product of two DG ideals
    • DGIdeal * DGModule (missing documentation)
    • DGIdeal * DGSubmodule (missing documentation)
    • DGIdeal ** Ring -- Base change of a DG ideal along a ring map
    • DGIdeal + DGIdeal -- Sum of two DG ideals
    • DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals
    • DGIdeal == DGIdeal -- Equality and containment of DG ideals
    • isSubset(DGIdeal,DGIdeal) -- see DGIdeal == DGIdeal -- Equality and containment of DG ideals
    • DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal
    • 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
    • DGModuleMap * DGModuleMap (missing documentation)
    • DGModuleMap * Number (missing documentation)
    • DGModuleMap * QQ (missing documentation)
    • DGModuleMap * RingElement (missing documentation)
    • DGModuleMap * ZZ (missing documentation)
    • DGModuleMap ** DGModuleMap -- Tensor product of DG module maps
    • DGModuleMap ** Ring -- Base change of a DG module map along a ring map
    • DGModuleMap + DGModuleMap (missing documentation)
    • DGModuleMap - DGModuleMap (missing documentation)
    • DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
    • isHomogeneous(DGModuleMap) -- see DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
    • isQuasiIsomorphism(DGModuleMap) -- see DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
    • map(DGModule,DGModule,ZZ) -- see DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
    • DGModuleMap == ZZ (missing documentation)
    • DGModuleMap Vector (missing documentation)
    • 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
    • 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
    • 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
    • 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
    • diff(DGAlgebra,RingElement) -- Apply the differential of a DGAlgebra to a ring element
    • diff(DGModule,Vector) -- Apply the DG module differential to an element
    • diff(DGQuotientModule,Vector) -- Apply the induced differential on a DG quotient module to an element
    • differential(DGAlgebra) -- see differential -- The list of generator differentials stored in a DG algebra or DG module
    • differential(DGModule) -- see differential -- The list of generator differentials stored in a DG algebra or DG module
    • directSum(DGModule) (missing documentation)
    • displayBlockDiff(DGAlgebra,Array,Array) -- see displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • displayBlockDiff(DGAlgebra,List,List) -- see displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • displayBlockDiff(DGAlgebra,VisibleList) -- see displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • displayBlockDiff(DGAlgebra,ZZ) -- see displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • 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(DGAlgebra) -- see ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
    • ensureDGAlgebraCaches(DGModule) -- see ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
    • expandGeomSeries(List,ZZ) -- see expandGeomSeries -- Expand a geometric series to a specified degree.
    • expandGeomSeries(RingElement,ZZ) -- see expandGeomSeries -- Expand a geometric series to a specified degree.
    • findNaryTrivialMasseyOperation(DGAlgebra,List,HashTable,ZZ) -- see findTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)
    • findTrivialMasseyOperation(DGAlgebra) -- see findTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)
    • freeDGAlgebra(Ring,List) -- see freeDGAlgebra -- Construct a free DG algebra with given generator degrees
    • freeDGModule(DGAlgebra,List) -- see freeDGModule -- Construct a free DG module over a DG algebra
    • generatorDegrees(DGAlgebra) -- see generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
    • generatorDegrees(DGModule) -- see generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
    • generators(DGAlgebra) (missing documentation)
    • generatorTable(DGModule) -- see generatorTable -- Display the generator list of a DG module with hom-degrees and differentials
    • getBasis(ZZ,DGAlgebra) -- see getBasis -- Get a basis for a particular homological degree of a DG algebra.
    • getBasis(ZZ,DGModule) -- Basis of the hom-degree-n piece of a DG module
    • getBasis(ZZ,Ring) -- Get a basis for a degree of a ring.
    • getBoundaryPreimage(DGAlgebra,List) -- see getBoundaryPreimage -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
    • getBoundaryPreimage(DGAlgebra,RingElement) -- see getBoundaryPreimage -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
    • 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
    • getDegNModule(ZZ,Ring,Ring) -- see getDegNModule -- Extract the degree-n strand of a graded algebra as a module over its degree-zero piece
    • getGenerators(DGAlgebra) -- see getGenerators -- Cycles whose classes generate the homology algebra of a DG algebra
    • HH DGAlgebra -- The homology algebra of a DG algebra
    • HH DGAlgebraMap -- The induced ring map on homology algebras
    • 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 DGModuleMap -- The induced map on graded homology modules over HH(A)
    • 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 DGAlgebra -- Computes the homology of a DG algebra as a module
    • HH_ZZ DGModule -- The degree-n homology of a DG module as a module over the base ring
    • HH_ZZ DGQuotientModule -- The degree-n homology of a DG quotient module
    • homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map
    • homologyAlgebra(DGAlgebra) -- see homologyAlgebra -- Compute the homology algebra of a DGAlgebra.
    • homologyClass(DGAlgebra,RingElement) -- see homologyClass -- Computes the element of the homology algebra corresponding to a cycle in a DGAlgebra.
    • homologyClass(DGModule,Vector) -- The homology class of a cycle in a DG module
    • homologyModule(DGAlgebra,Module) -- see homologyModule -- Compute the homology of a DGModule as a module over a DGAlgebra.
    • ideal(DGIdeal) (missing documentation)
    • identityDGAlgebraMap(DGAlgebra) -- see identityDGAlgebraMap -- The identity DGAlgebraMap on a DG algebra
    • identityDGModuleMap(DGModule) -- see identityDGModuleMap -- The identity DGModuleMap on a DG module
    • image(DGModuleMap) -- Image of a DG module map as a DG submodule of its target
    • intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals
    • intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient
    • invalidateDGAlgebraCache(DGAlgebra) -- see invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
    • invalidateDGAlgebraCache(DGModule) -- see invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
    • isAcyclic(DGAlgebra) -- see isAcyclic -- Determine whether a DG algebra has no higher homology
    • isAcyclic(DGModule) -- Determine whether a DG module has vanishing positive-degree homology
    • isAcyclic(DGQuotientModule) -- Determine whether a DG quotient module has vanishing positive-degree homology
    • isDGIdeal(DGAlgebra,Ideal) -- see isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A
    • 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?
    • isFreeDGModule(DGSubmodule) -- see isFreeDGModule -- Is the underlying A.natural-module free?
    • isGolod(Ring) -- see isGolod -- Determines if a ring is Golod
    • isGolodHomomorphism(QuotientRing) -- see isGolodHomomorphism -- Determines if the canonical map from the ambient ring is Golod
    • isHomogeneous(DGAlgebra) -- Determine if the DGAlgebra respects the gradings of the ring it is defined over.
    • isHomogeneous(DGIdeal) -- Predicates on a DG ideal
    • isZero(DGIdeal) -- see isHomogeneous(DGIdeal) -- Predicates on a DG ideal
    • isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
    • isHomogeneous(DGQuotientModule) -- see isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
    • isHomogeneous(DGSubmodule) -- see isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
    • isHomologyAlgebraTrivial(DGAlgebra) -- see isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial
    • isInjective(DGModuleMap) (missing documentation)
    • isIsomorphism(DGModuleMap) (missing documentation)
    • isMinimalSemifreeResolution(DGModule) -- see isMinimalSemifreeResolution -- Test whether a semifree DG module is minimal over its DG algebra
    • isQuasiIsomorphism(DGAlgebraMap) -- Determine whether a DG algebra map induces an isomorphism on homology
    • isSurjective(DGModuleMap) (missing documentation)
    • isValidDGAlgebra(DGAlgebra) -- see isValidDGAlgebra -- Structural invariant check for a DG algebra
    • isValidDGModule(DGModule) -- see isValidDGModule -- Structural-invariant check on a DG module
    • isWellDefined(DGAlgebra) -- Check that a DG algebra has correct structure and a differential that squares to zero
    • isWellDefined(DGIdeal) -- Verify that a DGIdeal has correct structure and is d-closed
    • isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero
    • isWellDefined(DGModuleMap) -- Check that a DG module map has correct structure and is a chain map
    • isWellDefined(DGQuotientModule) -- Check that a DG quotient module has a well-defined differential
    • isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module
    • isWellDefinedDifferential(DGAlgebra) -- see isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module
    • 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?
    • isZero(DGQuotientModule) -- see isZero(DGModule) -- Does the DG object have no natural generators?
    • isZero(DGSubmodule) -- see isZero(DGModule) -- Does the DG object have no natural generators?
    • kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source
    • killCycles(DGAlgebra) -- see killCycles -- Adjoin variables to kill non-bounding cycles in the lowest positive degree
    • killCycles(DGModule) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module
    • killHomologyAtDegree(DGAlgebra,ZZ) -- see killHomologyAtDegree -- Adjoin variables killing the homology of a DG algebra at a prescribed degree
    • koszulComplexDGA(Ring) -- see koszulComplexDGA -- The Koszul complex on the variables of a ring, as a DG algebra
    • koszulComplexDGA(Ideal) -- The Koszul complex on the generators of an ideal, as a DG algebra
    • koszulComplexDGA(List) -- The Koszul complex on a list of ring elements, as a DG algebra
    • koszulComplexDGM(DGAlgebra,Module) -- see koszulComplexDGM -- The Koszul complex of a module as a DG module
    • koszulComplexDGM(Module) -- see koszulComplexDGM -- The Koszul complex of a module as a DG module
    • liftToDGMap(DGAlgebra,DGAlgebra,RingMap) -- see liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
    • 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
    • masseyTripleProduct(DGAlgebra,RingElement,RingElement,RingElement) -- see masseyTripleProduct -- Computes the Massey triple product of a set of cycles or homology classes
    • masseyTripleProduct(DGAlgebra,ZZ,ZZ,ZZ) -- Computes the matrix representing all triple Massey operations.
    • maxDegree(DGAlgebra) -- see maxDegree -- Computes the maximum homological degree of a DGAlgebra
    • maxDegree(DGModule) -- Largest hom-degree present in a DG module
    • maxDegree(DGQuotientModule) -- Largest hom-degree present in a DG quotient module
    • generators(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
    • mingens(DGIdeal) -- Basic accessors on a DG ideal
    • module(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
    • numgens(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
    • ring(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
    • minimalModel(Ideal) -- see minimalModel -- Build the minimal DG algebra resolution of a quotient ring
    • minimalModel(QuotientRing) -- see minimalModel -- Build the minimal DG algebra resolution of a quotient ring
    • minimalSemifreeResolution(DGAlgebra,Module) -- see minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
    • minimalSemifreeResolution(Module) -- see minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
    • module(DGSubmodule) (missing documentation)
    • 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(DGAlgebra) -- Short-form display of a DG algebra
    • net(DGAlgebraMap) -- Short-form display of a DG algebra map
    • net(DGIdeal) (missing documentation)
    • net(DGModule) (missing documentation)
    • net(DGModuleMap) (missing documentation)
    • net(DGQuotientModule) (missing documentation)
    • net(DGSubmodule) (missing documentation)
    • Number * DGModuleMap (missing documentation)
    • numgens(DGAlgebra) (missing documentation)
    • numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
    • numgens(DGQuotientModule) -- see numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
    • numgens(DGSubmodule) -- see numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
    • polyDifferential(DGAlgebra,RingElement) -- see polyDifferential -- Matrix and element-level differentials of a DG algebra
    • polyDifferential(ZZ,DGAlgebra) -- see polyDifferential -- Matrix and element-level differentials of a DG algebra
    • polyDiffMonomial(DGAlgebra,RingElement) -- see polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra
    • minimalPresentation(DGAlgebra) -- see prune(DGAlgebra) -- Pruning a DG algebra is the identity
    • prune(DGAlgebra) -- Pruning a DG algebra is the identity
    • minimalPresentation(DGAlgebraMap) -- see prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
    • prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
    • minimalPresentation(DGIdeal) -- see prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
    • prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
    • minimalPresentation(DGModule) -- see prune(DGModule) -- Pruning a DG module is the identity
    • prune(DGModule) -- Pruning a DG module is the identity
    • minimalPresentation(DGModuleMap) -- see prune(DGModuleMap) -- Pruning a DG module map is the identity
    • prune(DGModuleMap) -- Pruning a DG module map is the identity
    • minimalPresentation(DGQuotientModule) -- see prune(DGQuotientModule) -- Minimize the presentation of a DG quotient module
    • prune(DGQuotientModule) -- Minimize the presentation of a DG 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
    • QQ * DGModuleMap (missing documentation)
    • rank(DGModule) -- Rank of the underlying free A.natural-module
    • rank(DGSubmodule) -- see rank(DGModule) -- Rank of the underlying free A.natural-module
    • restrictDifferential(DGAlgebra,List) -- see restrictDifferential -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
    • ring(DGAlgebra) (missing documentation)
    • ring(DGModule) (missing documentation)
    • ring(DGModuleMap) (missing documentation)
    • RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
    • ZZ % DGIdeal -- see RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
    • RingElement * DGModuleMap (missing documentation)
    • semifreeResolution(DGAlgebra,Module) -- see semifreeResolution -- Build a semifree DG module resolving a module over the base ring
    • semifreeResolution(Module) -- see semifreeResolution -- Build a semifree DG module resolving a module over the base ring
    • setDiff(DGAlgebra,List) -- see setDiff -- Install or replace the differential on a DG algebra
    • setDiff(DGModule,List) -- Install the list of generator differentials on a free DG module
    • source(DGAlgebraMap) -- The source of a DG algebra map
    • source(DGModuleMap) (missing documentation)
    • subDGAlgebra(DGAlgebra,List) -- see subDGAlgebra -- The sub-DG algebra generated by a chosen subset of DG generators
    • 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
    • super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
    • target(DGAlgebraMap) -- The target of a DG algebra map
    • target(DGModuleMap) (missing documentation)
    • toComplex(DGAlgebra) -- see toComplex -- Converts a DGAlgebra to a Complex
    • toComplex(DGAlgebra,ZZ) -- Converts a DGAlgebra to a Complex
    • toComplex(DGIdeal) -- Export a DG ideal to a Complex (not yet implemented)
    • 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
    • toComplex(DGQuotientModule) -- Export a DG quotient module to a Complex
    • toComplexMap(DGAlgebraMap) -- see toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap
    • toComplexMap(DGAlgebraMap,ZZ) -- see toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap
    • toComplexMap(DGModuleMap) -- Export a DG module map to a ComplexMap
    • toComplexMap(DGModuleMap,ZZ) -- The per-hom-degree component of a DG module map as a matrix
    • torAlgebra(Ring) -- see torAlgebra -- Computes the Tor algebra of a ring
    • torAlgebra(Ring,Ring) -- Computes Tor_R(S,k) up to a specified generating and relating degree.
    • torMap(RingMap) -- see torMap -- Compute the map of Tor algebras associated to a RingMap.
    • truncateGenerators(DGAlgebra,ZZ) -- see truncateGenerators -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
    • underlyingAlgebra(DGAlgebra) -- see underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
    • underlyingAlgebra(DGModule) -- see underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
    • underlyingRing(DGAlgebra) -- see underlyingRing -- The commutative base ring of a DG algebra or DG module
    • 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
    • zerothHomology(DGAlgebra) -- see zerothHomology -- The zeroth homology of a DG algebra as a ring
    • ZZ * DGModuleMap (missing documentation)
    • ZZ == DGModuleMap (missing documentation)
  • Symbols
    • AssertWellDefined -- Option to verify that a constructed DG algebra map respects differentials
    • cycles -- Chosen cycle representatives for homology generators of a DG algebra
    • dgAlgebra (missing documentation)
    • EndDegree -- Option to specify the last homological degree processed by an iterative DG construction
    • GenDegreeLimit -- Option to bound the homological degree searched for algebra generators
    • natural -- The underlying graded-commutative ring of a DGAlgebra or DGModule
    • RelDegreeLimit -- Option to bound the homological degree searched for algebra relations
    • ringMap -- The degree-zero ring map underlying a DGAlgebraMap
    • InitializeComplex -- see setDiff -- Install or replace the differential on a DG algebra
    • InitializeDegreeZeroHomology -- see setDiff -- Install or replace the differential on a DG algebra
    • StartDegree -- Option to specify the first homological degree processed by an iterative DG construction
    • TMOLimit -- Option to bound the arity of Massey operations searched for by findTrivialMasseyOperation

For the programmer

The object DGAlgebras is a package, defined in DGAlgebras.m2, with auxiliary files in DGAlgebras/.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:32:0.