DGAlgebras : Table of Contents

• DGAlgebras -- Data types and basic functions on differential graded (DG) Algebras.
  • Basic operations on DG Algebras -- Outlines some basic operations on DG Algebras
  • The Koszul complex as a DG Algebra -- an example
  • Basic operations on DG Algebra Maps -- Outlines some basic operations on DGAlgebraMaps
  • Basic operations on DG Module Maps -- Outlines some basic operations on DGModuleMaps
  • Base change and tensor with non-DG types -- Tensoring DG objects with ordinary rings and modules
    • DGAlgebra ** Ring -- Base change of a DG algebra to another ring
    • DGModule ** Ring -- Base change of a DG module along a ring map
    • DGSubmodule ** Ring -- Base change of a DG submodule along a ring map
    • DGQuotientModule ** Ring -- Base change of a DG quotient module along a ring map
    • DGIdeal ** Ring -- Base change of a DG ideal along a ring map
    • DGAlgebraMap ** Ring -- Base change of a DG algebra map along a ring map
    • DGModuleMap ** Ring -- Base change of a DG module map along a ring map
    • DGModule ** Module -- Exterior tensor product of a DG module with an ordinary free module
    • DGSubmodule ** Module -- Exterior tensor product of a DG submodule with an ordinary free module
    • DGQuotientModule ** Module -- Exterior tensor product of a DG quotient module with an ordinary free module
  • Operations on DG Ideals -- Sum, product, power, intersection, colon, and other ideal-algebra operations on DG ideals
    • 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
    • isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A
    • isWellDefined(DGIdeal) -- Verify that a DGIdeal has correct structure and is d-closed
    • DGIdeal + DGIdeal -- Sum of two DG ideals
    • DGIdeal * DGIdeal -- Product of two DG ideals
    • DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal
    • intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals
    • DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals
    • DGIdeal == DGIdeal -- Equality and containment of DG ideals
    • RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
    • mingens(DGIdeal) -- Basic accessors on a DG ideal
    • prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
    • DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal
    • isHomogeneous(DGIdeal) -- Predicates on a DG ideal
  • Building DG modules, submodules, and quotients -- The DGModule / DGSubmodule / DGQuotientModule hierarchy and its constructors
    • DGModule -- The class of DG modules over a DG algebra
    • DGSubmodule -- The class of d-closed submodules of a DG module
    • DGQuotientModule -- The class of quotient DG modules M / S
    • freeDGModule -- Construct a free DG module over a DG algebra
    • dgSubmodule -- Construct a d-closed submodule of a DG module
    • dgQuotientModule -- Construct the quotient DG module M / S
    • DGModule / DGSubmodule -- Quotient DG module: M / S
  • Module-like operations on DG modules -- Basic queries (generators, rank, degrees, homogeneity) extended to DG modules and their sub- and quotient modules
    • numgens(DGModule) -- Number of natural generators of a DG module (or sub- or quotient module)
    • rank(DGModule) -- Rank of the underlying free A.natural-module
    • degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
    • isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous
    • isFreeDGModule -- Is the underlying A.natural-module free?
    • isZero(DGModule) -- Does the DG object have no natural generators?
    • super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module
  • Operations on DG Submodules -- Sum, intersection, equality, and containment of DG submodules
    • DGSubmodule + DGSubmodule -- Sum of two DG submodules of a common ambient
    • intersect(DGSubmodule,DGSubmodule) -- Intersection of two DG submodules of a common ambient
    • DGSubmodule == DGSubmodule -- Equality and containment of DG submodules
  • Image, kernel, and cokernel of DG module maps -- Building DG submodules and DG quotient modules from a DG module map
    • image(DGModuleMap) -- Image of a DG module map as a DG submodule of its target
    • kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source
    • cokernel(DGModuleMap) -- Cokernel of a DG module map as a DG quotient module of its target
  • 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
    • HH_ZZ DGModule -- The degree-n homology of a DG module as a module over the base ring
    • HH DGModule -- The graded homology of a DG module as a module over HH(A)
    • homology(DGModuleMap,ZZ) -- The induced map on degree-n homology of a DG module map
    • HH_ZZ DGQuotientModule -- The degree-n homology of a DG quotient module
    • HH DGQuotientModule -- The graded homology of a DG quotient module
    • homologyClass(DGModule,Vector) -- The homology class of a cycle in a DG module
  • Pruning DG modules, submodules, quotients, and maps -- Minimizing presentations of DG types via prune and minimalPresentation
    • prune(DGSubmodule) -- Trim a DG submodule to a minimal generating set of its inclusion
    • prune(DGQuotientModule) -- Minimize the presentation of a DG quotient module
    • prune(DGModule) -- Pruning a DG module is the identity
    • prune(DGAlgebra) -- Pruning a DG algebra is the identity
    • prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
    • prune(DGModuleMap) -- Pruning a DG module map is the identity
  • Well-definedness, acyclicity, and quasi-isomorphism -- Predicates that check DG-theoretic conditions on DG types
    • isWellDefined(DGAlgebra) -- Check that a DG algebra has correct structure and a differential that squares to zero
    • isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero
    • isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module
    • isWellDefined(DGQuotientModule) -- Check that a DG quotient module has a well-defined differential
    • isWellDefined(DGModuleMap) -- Check that a DG module map has correct structure and is a chain map
    • 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
    • isQuasiIsomorphism(DGAlgebraMap) -- Determine whether a DG algebra map induces an isomorphism on homology
    • annihilator(DGSubmodule) -- The DG ideal of A annihilating a DG submodule
    • annihilator(DGQuotientModule) -- The DG ideal of A annihilating a DG quotient module
  • Semifree resolutions of DG modules -- Building semifree DG module resolutions via koszulComplexDGM, adjoinGenerators, killCycles, and semifreeResolution
    • koszulComplexDGM -- The Koszul complex of a module as a DG module
    • adjoinGenerators -- Adjoin new free generators to a DG module with prescribed differentials
    • killCycles(DGModule) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module
    • semifreeResolution -- Build a semifree DG module resolving a module over the base ring
    • semifreeResolution(...,StartDegree=>...) -- Option to specify the degree at which semifreeResolution starts killing cycles
    • semifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which semifreeResolution kills cycles
    • minimalSemifreeResolution -- Build a minimal semifree DG module resolution over a DG algebra
    • minimalSemifreeResolution(...,StartDegree=>...) -- Option to specify the degree at which minimalSemifreeResolution starts killing cycles
    • minimalSemifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which minimalSemifreeResolution kills cycles
    • isMinimalSemifreeResolution -- Test whether a semifree DG module is minimal over its DG algebra
  • Computing module differentials and visualizing DG modules -- Differential matrices, element-wise differentials, and visualization helpers for DG modules
    • moduleDifferential -- The hom-degree-n differential of a DG module as a matrix over the base ring
    • 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
    • generatorTable -- Display the generator list of a DG module with hom-degrees and differentials
    • dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module
    • moduleBlockDiff -- The hom-degree-n differential of a DG module as a labeled block matrix
    • displayModuleBlockDiff -- Pretty-print the labeled block matrix of a DG module differential
  • Building DG algebras from existing DG algebras -- Base change, generator restriction, truncation, and targeted cycle-killing for DG algebras
    • baseChange -- Transport a DG algebra along a ring map on its base
    • subDGAlgebra -- The sub-DG algebra generated by a chosen subset of DG generators
    • restrictDifferential -- Synonym for subDGAlgebra, named to match the restricted-differential vocabulary
    • truncateGenerators -- Restrict a DG algebra to its generators of hom-degree strictly above a bound
    • killHomologyAtDegree -- Adjoin variables killing the homology of a DG algebra at a prescribed degree
    • isValidDGAlgebra -- Structural invariant check for a DG algebra
    • isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module
  • Low-level differential computations and validity checks -- Primitives behind DG algebra and DG module differentials
    • polyDifferential -- Matrix and element-level differentials of a DG algebra
    • polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra
    • dgComplex -- Package a DG algebra or DG module as a Complex of free base-ring modules
    • isValidDGModule -- Structural-invariant check on a DG module
    • isDGSubmodule -- Test whether the image of an inclusion matrix is d-closed
    • getBoundaryPreimage(DGModule,Vector) -- Lift a boundary in a DG module to a preimage under the differential
    • getBoundaryPreimage(DGModule,List) -- Lift a list of boundaries sharing a hom-degree through the module differential
  • Accessors and cache management -- Reading state out of a DG algebra or DG module and maintaining its caches
    • underlyingRing -- The commutative base ring of a DG algebra or DG module
    • underlyingAlgebra -- The graded-commutative algebra carrying the DG structure
    • differential -- The list of generator differentials stored in a DG algebra or DG module
    • generatorDegrees -- The hom-degrees (and optional internal degrees) of the DG generators
    • ensureDGAlgebraCaches -- Guarantee that the standard cache sub-tables are present
    • invalidateDGAlgebraCache -- Discard cached derived data so that it is recomputed from scratch
  • DG module primitives and chain-complex export -- Installing differentials, reading degree data, and exporting to Complex
    • setDiff(DGModule,List) -- Install the list of generator differentials on a free DG module
    • maxDegree(DGModule) -- Largest hom-degree present in a DG module
    • maxDegree(DGQuotientModule) -- Largest hom-degree present in a DG quotient module
    • getBasis(ZZ,DGModule) -- Basis of the hom-degree-n piece of a DG module
    • 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
    • toComplex(DGIdeal) -- Export a DG ideal to a Complex (not yet implemented)
    • 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
    • displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
• 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(Ring) -- Tate's acyclic closure of the residue field, starting from the Koszul complex
• 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
• ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
• AssertWellDefined -- Option to verify that a constructed DG algebra map respects differentials
• blockDiff -- prepares a map for display
• cycles -- Chosen cycle representatives for homology generators of a DG algebra
• deviations -- Computes the deviations of the input ring, complex, or power series.
• deviations(...,DegreeLimit=>...) -- Maximum homological degree to which deviations are computed
• deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
• DGAlgebra -- The class of all DGAlgebras
• dgAlgebra (missing documentation)
• DGAlgebra ** DGAlgebra -- Exterior tensor product of DG algebras over a common ground ring
• 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
• dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
• DGModule ** DGModule -- Exterior tensor product of DG modules over DG algebras sharing a ground ring
• 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 == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps
• diff(DGAlgebra,RingElement) -- Apply the differential of a DGAlgebra to a ring element
• EndDegree -- Option to specify the last homological degree processed by an iterative DG construction
• expandGeomSeries -- Expand a geometric series to a specified degree.
• 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
• freeDGAlgebra -- Construct a free DG algebra with given generator degrees
• GenDegreeLimit -- Option to bound the homological degree searched for algebra generators
• 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,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.
• 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
• 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
• HH DGAlgebra -- The homology algebra of a DG algebra
• HH DGAlgebraMap -- The induced ring map on homology algebras
• HH DGModuleMap -- The induced map on graded homology modules over HH(A)
• HH_ZZ DGAlgebra -- Computes the homology of a DG algebra as a module
• 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
• 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
• isAcyclic -- Determine whether a DG algebra has no higher homology
• isAcyclic(...,EndDegree=>...) -- Option to specify the degree to finish checking acyclicity
• isGolod -- 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
• isHomogeneous(DGAlgebra) -- Determine if the DGAlgebra respects the gradings of the ring it is defined over.
• isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial
• isHomologyAlgebraTrivial(...,GenDegreeLimit=>...) -- Maximum homological degree of generators used by isHomologyAlgebraTrivial
• 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
• 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
• 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
• masseyTripleProduct(DGAlgebra,ZZ,ZZ,ZZ) -- Computes the matrix representing all triple Massey operations.
• maxDegree -- Computes the maximum homological degree of a DGAlgebra
• minimalModel -- Build the minimal DG algebra resolution of a quotient 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
• RelDegreeLimit -- Option to bound the homological degree searched for algebra relations
• ringMap -- The degree-zero ring map underlying a DGAlgebraMap
• setDiff -- Install or replace the differential on a DG algebra
• source(DGAlgebraMap) -- The source of a DG algebra map
• StartDegree -- Option to specify the first homological degree processed by an iterative DG construction
• target(DGAlgebraMap) -- The target of a DG algebra map
• TMOLimit -- Option to bound the arity of Massey operations searched for by findTrivialMasseyOperation
• toComplex -- Converts a DGAlgebra to a Complex
• toComplex(DGAlgebra,ZZ) -- Converts a DGAlgebra to a Complex
• toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap
• 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,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.
• Verbosity -- Option controlling the verbosity level of DGAlgebras computations
• zeroDGModuleMap -- The zero DGModuleMap between two DG modules over a common algebra
• zerothHomology -- The zeroth homology of a DG algebra as a ring