DGAlgebras : Index

• Accessors and cache management -- Reading state out of a DG algebra or DG module and maintaining its caches
• acyclicClosure -- Tate's construction of a semifree acyclic DG algebra extension
• acyclicClosure(...,EndDegree=>...) -- Option to specify the degree to stop computing the acyclic closure
• acyclicClosure(...,StartDegree=>...) -- Option to specify the degree to start computing the acyclic closure.
• acyclicClosure(...,Variable=>...) -- Option to name the DG algebra generators adjoined during acyclic closure
• acyclicClosure(DGAlgebra) -- 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 -- Adjoin new free generators to a DG module with prescribed differentials
• adjoinGenerators(DGModule,List) -- 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
• adjoinVariables(...,Variable=>...) -- Option to name the DG algebra generators adjoined by adjoinVariables
• adjoinVariables(DGAlgebra,List) -- Extend a DG algebra by a user-chosen list of cycles to be killed
• ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
• ambient(DGIdeal) -- Accessors for the ambient object, inclusion, projection, and killed submodule
• ambient(DGQuotientModule) -- Accessors for the ambient object, inclusion, projection, and killed submodule
• ambient(DGSubmodule) -- 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
• AssertWellDefined -- Option to verify that a constructed DG algebra map respects differentials
• Base change and tensor with non-DG types -- Tensoring DG objects with ordinary rings and modules
• baseChange -- Transport a DG algebra along a ring map on its base
• baseChange(...,InitializeComplex=>...) -- Transport a DG algebra along a ring map on its base
• baseChange(...,InitializeDegreeZeroHomology=>...) -- Transport a DG algebra along a ring map on its base
• baseChange(DGAlgebra,Ring) -- Transport a DG algebra along a ring map on its base
• baseChange(DGAlgebra,RingMap) -- Transport a DG algebra along a ring map on its base
• Basic operations on DG Algebra Maps -- Outlines some basic operations on DGAlgebraMaps
• Basic operations on DG Algebras -- Outlines some basic operations on DG Algebras
• Basic operations on DG Module Maps -- Outlines some basic operations on DGModuleMaps
• blockDiff -- prepares a map for display
• blockDiff(DGAlgebra,ZZ) -- prepares a map for display
• Building DG algebras from existing DG algebras -- Base change, generator restriction, truncation, and targeted cycle-killing for DG algebras
• Building DG modules, submodules, and quotients -- The DGModule / DGSubmodule / DGQuotientModule hierarchy and its constructors
• cokernel(DGModuleMap) -- Cokernel of a DG module map as a DG quotient module of its target
• Computing module differentials and visualizing DG modules -- Differential matrices, element-wise differentials, and visualization helpers for DG modules
• cover(DGQuotientModule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
• cycles -- Chosen cycle representatives for homology generators of a DG algebra
• degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
• degrees(DGQuotientModule) -- Multi-degrees of the natural generators of a DG module
• degrees(DGSubmodule) -- Multi-degrees of the natural generators of a DG module
• deviations -- Computes the deviations of the input ring, complex, or power series.
• deviations(...,DegreeLimit=>...) -- Maximum homological degree to which deviations are computed
• deviations(Complex) -- Computes the deviations of the input ring, complex, or power series.
• deviations(Ring) -- Computes the deviations of the input ring, complex, or power series.
• deviations(RingElement,List) -- Computes the deviations of the input ring, complex, or power series.
• deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
• deviationsToPoincare(...,DegreeLimit=>...) -- Computes the power series corresponding to a set of deviations.
• deviationsToPoincare(HashTable) -- Computes the power series corresponding to a set of deviations.
• DG module primitives and chain-complex export -- Installing differentials, reading degree data, and exporting to Complex
• DGAlgebra -- The class of all DGAlgebras
• dgAlgebra (missing documentation)
• 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 -- The class of DG algebra maps
• 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(DGAlgebra,DGAlgebra,Matrix) -- Define a DG algebra map between DG algebras.
• dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
• dgAlgebraMultMap(DGAlgebra,RingElement) -- Returns the chain map corresponding to multiplication by a cycle.
• DGAlgebras -- Data types and basic functions on differential graded (DG) Algebras.
• dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
• dgComplex(DGAlgebra) -- Package a DG algebra or DG module as a Complex of free base-ring modules
• dgComplex(DGModule) -- Package a DG algebra or DG module as a Complex of free base-ring modules
• DGIdeal -- The class of d-closed graded ideals of a DG algebra
• dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
• DGIdeal * DGIdeal -- Product of two DG ideals
• 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
• DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal
• dgIdeal(DGAlgebra,Ideal) -- Construct a DG ideal from a list, matrix, or ideal of generators
• dgIdeal(DGAlgebra,List) -- Construct a DG ideal from a list, matrix, or ideal of generators
• dgIdeal(DGAlgebra,Matrix) -- Construct a DG ideal from a list, matrix, or ideal of generators
• DGModule -- The class of DG modules over a DG algebra
• 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
• DGModule ** Ring -- Base change of a DG module along a ring map
• DGModule / DGSubmodule -- Quotient DG module: M / S
• DGModuleMap -- The class of morphisms of DG modules over a common DG algebra
• dgModuleMap -- Construct a DGModuleMap from a matrix or from a list of image Vectors
• DGModuleMap ** DGModuleMap -- Tensor product of DG module maps
• DGModuleMap ** Ring -- Base change of a DG module map along a ring map
• DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
• dgModuleMap(DGModule,DGModule,List) -- Construct a DGModuleMap from a matrix or from a list of image Vectors
• dgModuleMap(DGModule,DGModule,Matrix) -- 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
• dgModuleSummary(DGModule) -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
• dgModuleSummary(DGModule,ZZ) -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
• DGQuotientModule -- The class of quotient DG modules M / S
• dgQuotientModule -- Construct the quotient DG module M / S
• 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
• dgQuotientModule(DGModule,DGSubmodule) -- Construct the quotient DG module M / S
• DGSubmodule -- The class of d-closed submodules of a DG module
• dgSubmodule -- Construct a d-closed submodule of a DG module
• 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
• dgSubmodule(DGModule,List) -- Construct a d-closed submodule of a DG module
• dgSubmodule(DGModule,Matrix) -- Construct a d-closed submodule of a DG module
• 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 -- The list of generator differentials stored in a DG algebra or DG module
• differential(DGAlgebra) -- The list of generator differentials stored in a DG algebra or DG module
• differential(DGModule) -- The list of generator differentials stored in a DG algebra or DG module
• displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
• displayBlockDiff(DGAlgebra,Array,Array) -- Shows natural decomposition of a map in the Tate resolution
• displayBlockDiff(DGAlgebra,List,List) -- Shows natural decomposition of a map in the Tate resolution
• displayBlockDiff(DGAlgebra,VisibleList) -- Shows natural decomposition of a map in the Tate resolution
• displayBlockDiff(DGAlgebra,ZZ) -- Shows natural decomposition of a map in the Tate resolution
• displayModuleBlockDiff -- Pretty-print the labeled block matrix of a DG module differential
• displayModuleBlockDiff(DGModule,List,List) -- Pretty-print the labeled block matrix of a DG module differential
• displayModuleBlockDiff(DGModule,ZZ) -- Pretty-print the labeled block matrix of a DG module differential
• EndDegree -- Option to specify the last homological degree processed by an iterative DG construction
• ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
• ensureDGAlgebraCaches(DGAlgebra) -- Guarantee that the standard cache sub-tables are present
• ensureDGAlgebraCaches(DGModule) -- Guarantee that the standard cache sub-tables are present
• expandGeomSeries -- Expand a geometric series to a specified degree.
• expandGeomSeries(List,ZZ) -- Expand a geometric series to a specified degree.
• expandGeomSeries(RingElement,ZZ) -- Expand a geometric series to a specified degree.
• findNaryTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)
• findNaryTrivialMasseyOperation(DGAlgebra,List,HashTable,ZZ) -- 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)
• findTrivialMasseyOperation(...,GenDegreeLimit=>...) -- Maximum homological degree of generators used in the trivial Massey operation search
• findTrivialMasseyOperation(...,TMOLimit=>...) -- Maximum arity of Massey products searched for bounding
• findTrivialMasseyOperation(DGAlgebra) -- Finds a trivial Massey operation on a set of generators of H(A)
• freeDGAlgebra -- Construct a free DG algebra with given generator degrees
• freeDGAlgebra(...,Variable=>...) -- Construct a free DG algebra with given generator degrees
• freeDGAlgebra(Ring,List) -- Construct a free DG algebra with given generator degrees
• freeDGModule -- Construct a free DG module over a DG algebra
• freeDGModule(DGAlgebra,List) -- Construct a free DG module over a DG algebra
• GenDegreeLimit -- Option to bound the homological degree searched for algebra generators
• generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
• generatorDegrees(DGAlgebra) -- The hom-degrees (and optional internal degrees) of the DG generators
• generatorDegrees(DGModule) -- The hom-degrees (and optional internal degrees) of the DG generators
• generators(DGIdeal) -- Basic accessors on a DG ideal
• generatorTable -- Display the generator list of a DG module with hom-degrees and differentials
• generatorTable(DGModule) -- 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.
• getBasis(...,Limit=>...) -- Maximum number of basis elements to return from getBasis
• getBasis(ZZ,DGAlgebra) -- 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 -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
• getBoundaryPreimage(DGAlgebra,List) -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
• getBoundaryPreimage(DGAlgebra,RingElement) -- 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 -- Extract the degree-n strand of a graded algebra as a module over its degree-zero piece
• getDegNModule(ZZ,Ring,Ring) -- 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
• getGenerators(...,DegreeLimit=>...) -- Last homological degree in which to search for algebra generators
• getGenerators(...,StartDegree=>...) -- First homological degree in which to search for new algebra generators
• getGenerators(...,Verbosity=>...) -- Option to control progress output from getGenerators
• getGenerators(DGAlgebra) -- 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)
• HH DGModuleMap -- The induced map on graded homology modules over HH(A)
• 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 of DG modules and DG module maps -- Computing H_n and H_* for DG modules, DG submodules, DG quotient modules, and DG module maps
• homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map
• homologyAlgebra -- Compute the homology algebra of a DGAlgebra.
• homologyAlgebra(...,GenDegreeLimit=>...) -- Maximum homological degree searched for homology-algebra generators
• homologyAlgebra(...,RelDegreeLimit=>...) -- Maximum homological degree searched for homology-algebra relations
• homologyAlgebra(...,Verbosity=>...) -- Option to control progress output from homologyAlgebra
• homologyAlgebra(DGAlgebra) -- Compute the homology algebra of a DGAlgebra.
• homologyClass -- Computes the element of the homology algebra corresponding to a cycle in a DGAlgebra.
• homologyClass(DGAlgebra,RingElement) -- 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 -- Compute the homology of a DGModule as a module over a DGAlgebra.
• homologyModule(DGAlgebra,Module) -- Compute the homology of a DGModule as a module over a DGAlgebra.
• homologyModule(DGModule) -- The graded homology of a DG module as a module over HH(A)
• homologyModule(DGQuotientModule) -- The graded homology of a DG quotient module
• identityDGAlgebraMap -- The identity DGAlgebraMap on a DG algebra
• identityDGAlgebraMap(DGAlgebra) -- The identity DGAlgebraMap on a DG algebra
• identityDGModuleMap -- The identity DGModuleMap on a DG module
• identityDGModuleMap(DGModule) -- The identity DGModuleMap on a DG module
• image(DGModuleMap) -- Image of a DG module map as a DG submodule of its target
• Image, kernel, and cokernel of DG module maps -- Building DG submodules and DG quotient modules from a DG module map
• inclusion -- Accessors for the ambient object, inclusion, projection, and killed submodule
• inclusion(DGSubmodule) -- Accessors for the ambient object, inclusion, projection, and killed submodule
• InitializeComplex -- Install or replace the differential on a DG algebra
• InitializeDegreeZeroHomology -- Install or replace the differential on a DG algebra
• intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals
• intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient
• invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
• invalidateDGAlgebraCache(DGAlgebra) -- Discard cached derived data so that it is recomputed from scratch
• invalidateDGAlgebraCache(DGModule) -- Discard cached derived data so that it is recomputed from scratch
• isAcyclic -- Determine whether a DG algebra has no higher homology
• isAcyclic(...,EndDegree=>...) -- Option to specify the degree to finish checking acyclicity
• isAcyclic(DGAlgebra) -- Determine whether a DG algebra has no higher homology
• isAcyclic(DGModule) -- Determine whether a DG module has vanishing positive-degree homology
• isAcyclic(DGModule,EndDegree=>...) -- Determine whether a DG module has vanishing positive-degree homology
• isAcyclic(DGQuotientModule) -- Determine whether a DG quotient module has vanishing positive-degree homology
• isAcyclic(DGQuotientModule,EndDegree=>...) -- Determine whether a DG quotient module has vanishing positive-degree homology
• isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A
• isDGIdeal(DGAlgebra,Ideal) -- 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
• isDGSubmodule(DGModule,Matrix) -- Test whether the image of an inclusion matrix is d-closed
• isFreeDGModule -- Is the underlying A.natural-module free?
• isFreeDGModule(DGModule) -- Is the underlying A.natural-module free?
• isFreeDGModule(DGSubmodule) -- Is the underlying A.natural-module free?
• isGolod -- Determines if a ring is Golod
• isGolod(Ring) -- Determines if a ring is Golod
• isGolodHomomorphism -- Determines if the canonical map from the ambient ring is Golod
• isGolodHomomorphism(...,GenDegreeLimit=>...) -- Maximum homological degree of generators used by isGolodHomomorphism
• isGolodHomomorphism(...,TMOLimit=>...) -- Option to cap the arity of the trivial Massey operation used in isGolodHomomorphism
• isGolodHomomorphism(QuotientRing) -- 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
• isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
• isHomogeneous(DGModuleMap) -- Elementary predicates and constructors for DGModuleMaps
• isHomogeneous(DGQuotientModule) -- Test whether the underlying graded module is homogeneous
• isHomogeneous(DGSubmodule) -- Test whether the underlying graded module is homogeneous
• isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial
• isHomologyAlgebraTrivial(...,GenDegreeLimit=>...) -- Maximum homological degree of generators used by isHomologyAlgebraTrivial
• isHomologyAlgebraTrivial(DGAlgebra) -- Determines if the homology algebra of a DGAlgebra is trivial
• isMinimalSemifreeResolution -- Test whether a semifree DG module is minimal over its DG algebra
• isMinimalSemifreeResolution(DGModule) -- 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
• isQuasiIsomorphism(DGModuleMap) -- Elementary predicates and constructors for DGModuleMaps
• isSubset(DGIdeal,DGIdeal) -- Equality and containment of DG ideals
• isSubset(DGSubmodule,DGSubmodule) -- Equality and containment of DG submodules
• isValidDGAlgebra -- Structural invariant check for a DG algebra
• isValidDGAlgebra(DGAlgebra) -- Structural invariant check for a DG algebra
• isValidDGModule -- Structural-invariant check on a DG module
• isValidDGModule(DGModule) -- 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(DGAlgebraMap) -- Define a DG algebra map between DG algebras.
• 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 -- Semantic check that d^2 = 0 for a DG algebra or DG module
• isWellDefinedDifferential(DGAlgebra) -- Semantic check that d^2 = 0 for a DG algebra or DG module
• isWellDefinedDifferential(DGModule) -- Semantic check that d^2 = 0 for a DG algebra or DG module
• isZero -- Does the DG object have no natural generators?
• isZero(DGIdeal) -- Predicates on a DG ideal
• isZero(DGModule) -- Does the DG object have no natural generators?
• isZero(DGQuotientModule) -- Does the DG object have no natural generators?
• isZero(DGSubmodule) -- Does the DG object have no natural generators?
• kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source
• killCycles -- Adjoin variables to kill non-bounding cycles in the lowest positive degree
• killCycles(...,EndDegree=>...) -- Last homological degree processed by killCycles
• killCycles(...,StartDegree=>...) -- First homological degree from which killCycles begins searching for cycles
• killCycles(...,Variable=>...) -- Option to name the DG algebra generators adjoined by killCycles
• killCycles(DGAlgebra) -- 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
• killCycles(DGModule,EndDegree=>...) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module
• killCycles(DGModule,StartDegree=>...) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module
• killHomologyAtDegree -- Adjoin variables killing the homology of a DG algebra at a prescribed degree
• killHomologyAtDegree(DGAlgebra,ZZ) -- 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
• koszulComplexDGA(...,Variable=>...) -- 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
• koszulComplexDGA(Ring) -- The Koszul complex on the variables of a ring, as a DG algebra
• koszulComplexDGM -- The Koszul complex of a module as a DG module
• koszulComplexDGM(DGAlgebra,Module) -- The Koszul complex of a module as a DG module
• koszulComplexDGM(Module) -- The Koszul complex of a module as a DG module
• liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
• liftToDGMap(...,EndDegree=>...) -- Lift a ring homomorphism in degree zero to a DG algebra morphism
• liftToDGMap(DGAlgebra,DGAlgebra,RingMap) -- 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
• liftToDGModuleMap(...,EndDegree=>...) -- Lift an image of hom-degree-0 generators to a full DGModuleMap
• liftToDGModuleMap(DGModule,DGModule,List) -- Lift an image of hom-degree-0 generators to a full DGModuleMap
• liftToDGModuleMap(DGModule,DGModule,Matrix) -- Lift an image of hom-degree-0 generators to a full DGModuleMap
• liftToDGModuleMap(DGModule,DGModule,Vector) -- Lift an image of hom-degree-0 generators to a full DGModuleMap
• Low-level differential computations and validity checks -- Primitives behind DG algebra and DG module differentials
• map(DGModule,DGModule,ZZ) -- Elementary predicates and constructors for DGModuleMaps
• masseyTripleProduct -- Computes the Massey triple product of a set of cycles or homology classes
• masseyTripleProduct(DGAlgebra,RingElement,RingElement,RingElement) -- 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 -- Computes the maximum homological degree of a DGAlgebra
• maxDegree(DGAlgebra) -- 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
• mingens(DGIdeal) -- Basic accessors on a DG ideal
• minimalModel -- Build the minimal DG algebra resolution of a quotient ring
• minimalModel(...,EndDegree=>...) -- Build the minimal DG algebra resolution of a quotient ring
• minimalModel(...,StartDegree=>...) -- Build the minimal DG algebra resolution of a quotient ring
• minimalModel(...,Variable=>...) -- Build the minimal DG algebra resolution of a quotient ring
• minimalModel(Ideal) -- Build the minimal DG algebra resolution of a quotient ring
• minimalModel(QuotientRing) -- Build the minimal DG algebra resolution of a quotient ring
• minimalPresentation(DGAlgebra) -- Pruning a DG algebra is the identity
• minimalPresentation(DGAlgebraMap) -- Pruning a DG algebra map is the identity
• minimalPresentation(DGIdeal) -- Trim a DG ideal down to a minimal generating set
• minimalPresentation(DGModule) -- Pruning a DG module is the identity
• minimalPresentation(DGModuleMap) -- Pruning a DG module map is the identity
• minimalPresentation(DGQuotientModule) -- Minimize the presentation of a DG quotient module
• minimalPresentation(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion
• minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
• minimalSemifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which minimalSemifreeResolution kills cycles
• minimalSemifreeResolution(...,StartDegree=>...) -- Option to specify the degree at which minimalSemifreeResolution starts killing cycles
• minimalSemifreeResolution(DGAlgebra,Module) -- Build a minimal semifree DG module resolution over a DG algebra
• minimalSemifreeResolution(Module) -- Build a minimal semifree DG module resolution over a DG algebra
• Module ** DGModule -- Exterior tensor product of a DG module with an ordinary free module
• Module ** DGQuotientModule -- Exterior tensor product of a DG quotient module with an ordinary free module
• Module ** DGSubmodule -- Exterior tensor product of a DG submodule with an ordinary free module
• module(DGIdeal) -- Basic accessors on a DG ideal
• Module-like operations on DG modules -- Basic queries (generators, rank, degrees, homogeneity) extended to DG modules and their sub- and quotient modules
• moduleBlockDiff -- The hom-degree-n differential of a DG module as a labeled block matrix
• moduleBlockDiff(DGModule,ZZ) -- 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
• moduleDifferential(ZZ,DGModule) -- The hom-degree-n differential of a DG module as a matrix over the base ring
• natural -- The underlying graded-commutative ring of a DGAlgebra or DGModule
• net(DGAlgebra) -- Short-form display of a DG algebra
• net(DGAlgebraMap) -- Short-form display of a DG algebra map
• numgens(DGIdeal) -- Basic accessors on a DG ideal
• numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
• numgens(DGQuotientModule) -- Number of natural generators of a DG module (or sub- or quotient module)
• numgens(DGSubmodule) -- Number of natural generators of a DG module (or sub- or quotient module)
• Operations on DG Ideals -- Sum, product, power, intersection, colon, and other ideal-algebra operations on DG ideals
• Operations on DG Submodules -- Sum, intersection, equality, and containment of DG submodules
• polyDifferential -- Matrix and element-level differentials of a DG algebra
• polyDifferential(DGAlgebra,RingElement) -- Matrix and element-level differentials of a DG algebra
• polyDifferential(ZZ,DGAlgebra) -- Matrix and element-level differentials of a DG algebra
• polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra
• polyDiffMonomial(DGAlgebra,RingElement) -- Leibniz rule on a single monomial of a DG algebra
• projection -- Accessors for the ambient object, inclusion, projection, and killed submodule
• projection(DGQuotientModule) -- Accessors for the ambient object, inclusion, projection, and killed submodule
• prune(DGAlgebra) -- Pruning a DG algebra is the identity
• prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
• prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
• prune(DGModule) -- Pruning a DG module is the identity
• prune(DGModuleMap) -- Pruning a DG module map is the identity
• prune(DGQuotientModule) -- Minimize the presentation of a DG quotient module
• prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion
• Pruning DG modules, submodules, quotients, and maps -- Minimizing presentations of DG types via prune and minimalPresentation
• rank(DGModule) -- Rank of the underlying free A.natural-module
• rank(DGSubmodule) -- Rank of the underlying free A.natural-module
• relations(DGQuotientModule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
• RelDegreeLimit -- Option to bound the homological degree searched for algebra relations
• restrictDifferential -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
• restrictDifferential(...,InitializeComplex=>...) -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
• restrictDifferential(...,InitializeDegreeZeroHomology=>...) -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
• restrictDifferential(DGAlgebra,List) -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
• ring(DGIdeal) -- Basic accessors on a DG ideal
• RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
• ringMap -- The degree-zero ring map underlying a DGAlgebraMap
• Semifree resolutions of DG modules -- Building semifree DG module resolutions via koszulComplexDGM, adjoinGenerators, killCycles, and semifreeResolution
• semifreeResolution -- Build a semifree DG module resolving a module over the base ring
• semifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which semifreeResolution kills cycles
• semifreeResolution(...,StartDegree=>...) -- Option to specify the degree at which semifreeResolution starts killing cycles
• semifreeResolution(DGAlgebra,Module) -- Build a semifree DG module resolving a module over the base ring
• semifreeResolution(Module) -- Build a semifree DG module resolving a module over the base ring
• setDiff -- Install or replace the differential on a DG algebra
• setDiff(...,InitializeComplex=>...) -- Install or replace the differential on a DG algebra
• setDiff(...,InitializeDegreeZeroHomology=>...) -- Install or replace the differential on a DG algebra
• setDiff(DGAlgebra,List) -- 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
• StartDegree -- Option to specify the first homological degree processed by an iterative DG construction
• subDGAlgebra -- The sub-DG algebra generated by a chosen subset of DG generators
• subDGAlgebra(...,InitializeComplex=>...) -- The sub-DG algebra generated by a chosen subset of DG generators
• subDGAlgebra(...,InitializeDegreeZeroHomology=>...) -- The sub-DG algebra generated by a chosen subset of DG generators
• subDGAlgebra(DGAlgebra,List) -- The sub-DG algebra generated by a chosen subset of DG generators
• subDGModule -- Accessors for the ambient object, inclusion, projection, and killed submodule
• subDGModule(DGQuotientModule) -- Accessors for the ambient object, inclusion, projection, and killed submodule
• super(DGQuotientModule) -- 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
• tensorFactors -- Exterior tensor product of DG algebras over a common ground ring
• tensorFactors(DGAlgebra) -- Exterior tensor product of DG algebras over a common ground ring
• tensorInclusions -- Exterior tensor product of DG algebras over a common ground ring
• tensorInclusions(DGAlgebra) -- Exterior tensor product of DG algebras over a common ground ring
• The Koszul complex as a DG Algebra -- an example
• TMOLimit -- Option to bound the arity of Massey operations searched for by findTrivialMasseyOperation
• toComplex -- Converts a DGAlgebra to a Complex
• toComplex(DGAlgebra) -- 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 -- Construct the ComplexMap associated to a DGAlgebraMap
• toComplexMap(...,AssertWellDefined=>...) -- Construct the ComplexMap associated to a DGAlgebraMap
• toComplexMap(...,EndDegree=>...) -- Construct the ComplexMap associated to a DGAlgebraMap
• toComplexMap(DGAlgebraMap) -- Construct the ComplexMap associated to a DGAlgebraMap
• toComplexMap(DGAlgebraMap,ZZ) -- 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 -- Computes the Tor algebra of a ring
• torAlgebra(...,GenDegreeLimit=>...) -- Maximum homological degree searched for Tor-algebra generators
• torAlgebra(...,RelDegreeLimit=>...) -- Maximum homological degree searched for Tor-algebra relations
• torAlgebra(Ring) -- Computes the Tor algebra of a ring
• torAlgebra(Ring,Ring) -- Computes Tor_R(S,k) up to a specified generating and relating degree.
• torMap -- Compute the map of Tor algebras associated to a RingMap.
• torMap(...,GenDegreeLimit=>...) -- Compute the map of Tor algebras associated to a RingMap.
• torMap(RingMap) -- 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
• truncateGenerators(...,InitializeComplex=>...) -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
• truncateGenerators(...,InitializeDegreeZeroHomology=>...) -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
• truncateGenerators(DGAlgebra,ZZ) -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
• underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
• underlyingAlgebra(DGAlgebra) -- The graded-commutative algebra carrying the DG structure
• underlyingAlgebra(DGModule) -- The graded-commutative algebra carrying the DG structure
• underlyingRing -- The commutative base ring of a DG algebra or DG module
• underlyingRing(DGAlgebra) -- The commutative base ring of a DG algebra or DG module
• underlyingRing(DGModule) -- The commutative base ring of a DG algebra or DG module
• Verbosity -- Option controlling the verbosity level of DGAlgebras computations
• Well-definedness, acyclicity, and quasi-isomorphism -- Predicates that check DG-theoretic conditions on DG types
• zeroDGModuleMap -- The zero DGModuleMap between two DG modules over a common algebra
• zeroDGModuleMap(DGModule,DGModule) -- The zero DGModuleMap between two DG modules over a common algebra
• zerothHomology -- The zeroth homology of a DG algebra as a ring
• zerothHomology(DGAlgebra) -- The zeroth homology of a DG algebra as a ring
• ZZ % DGIdeal -- Reduce a ring element modulo a DG ideal