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DGAlgebras :: DGAlgebras

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

Description

This package is used to define and manipulate DG algebras.

Author

Version

This documentation describes version 1.0.1 of DGAlgebras.

Source code

The source code from which this documentation is derived is in the file DGAlgebras.m2.

Exports

  • Types
  • Functions and commands
    • acyclicClosure -- Compute theae acyclic closure of a DGAlgebra.
    • adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
    • deviations -- Computes the deviations of the input ring, complex, or power series.
    • deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
    • dgAlgebraMap -- Define a DG algebra map between DG algebras.
    • 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 -- Constructs a DGAlgebra
    • getBasis -- Get a basis for a particular homological degree of a DG algebra.
    • getDegNModule -- Compute a presentation of M_i as an R-module
    • getGenerators -- Returns a list of cycles whose images generate HH(A) as an algebra
    • homologyAlgebra -- Compute the homology algebra of a DGAlgebra.
    • isAcyclic -- Determines if a DGAlgebra is acyclic.
    • 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
    • killCycles -- Adjoins variables to make non-bounding cycles boundaries in the lowest positive degree with nontrivial homology.
    • koszulComplexDGA -- Returns the Koszul complex as a DGAlgebra
    • liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
    • maxDegree -- Computes the maximum homological degree of a DGAlgebra
    • setDiff -- Sets the differential of a DGAlgebra manually.
    • toComplex -- Converts a DGAlgebra to a ChainComplex
    • toComplexMap -- Construct the ChainComplexMap associated to a DGAlgebraMap
    • torAlgebra -- Computes the Tor algebra of a ring
    • torMap -- Compute the map of Tor algebras associated to a RingMap.
    • zerothHomology -- Compute the zeroth homology of the DGAlgebra A as a ring.
  • Methods
    • acyclicClosure(DGAlgebra), see acyclicClosure -- Compute theae acyclic closure of a DGAlgebra.
    • adjoinVariables(DGAlgebra,List), see adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
    • DGAlgebra ** DGAlgebra -- Tensor product of a DGAlgebra and another ring.
    • DGAlgebra ** Ring -- Tensor product of a DGAlgebra and another ring.
    • 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.
    • 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)
    • getBasis(ZZ,DGAlgebra), see getBasis -- Get a basis for a particular homological degree of a DG algebra.
    • getGenerators(DGAlgebra), see getGenerators -- Returns a list of cycles whose images generate HH(A) as an algebra
    • HH DGAlgebra -- Compute the homology algebra of a DGAlgebra.
    • HH DGAlgebraMap -- Computes the homomorphism in homology associated to a DGAlgebraMap.
    • homology(DGAlgebraMap,ZZ), see HH DGAlgebraMap -- Computes the homomorphism in homology associated to a DGAlgebraMap.
    • HH_ZZ DGAlgebra -- Computes the homology of a DG algebra as a module
    • homologyAlgebra(DGAlgebra), see homologyAlgebra -- Compute the homology algebra of a DGAlgebra.
    • isAcyclic(DGAlgebra), see isAcyclic -- Determines if a DGAlgebra is acyclic.
    • isHomogeneous(DGAlgebra) -- Determine if the DGAlgebra respects the gradings of the ring it is defined over.
    • isHomologyAlgebraTrivial(DGAlgebra), see isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial
    • killCycles(DGAlgebra), see killCycles -- Adjoins variables to make non-bounding cycles boundaries in the lowest positive degree with nontrivial homology.
    • liftToDGMap(DGAlgebra,DGAlgebra,RingMap), see liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
    • maxDegree(DGAlgebra), see maxDegree -- Computes the maximum homological degree of a DGAlgebra
    • net(DGAlgebra) -- Outputs the pertinent information about a DGAlgebra
    • net(DGAlgebraMap) -- Outputs the pertinent information about a DGAlgebraMap
    • setDiff(DGAlgebra,List), see setDiff -- Sets the differential of a DGAlgebra manually.
    • source(DGAlgebraMap) -- Outputs the source of a DGAlgebraMap
    • target(DGAlgebraMap) -- Outputs the target of a DGAlgebraMap
    • toComplex(DGAlgebra), see toComplex -- Converts a DGAlgebra to a ChainComplex
    • toComplex(DGAlgebra,ZZ) -- Converts a DGAlgebra to a ChainComplex
    • toComplexMap(DGAlgebraMap), see toComplexMap -- Construct the ChainComplexMap associated to a DGAlgebraMap
    • toComplexMap(DGAlgebraMap,ZZ), see toComplexMap -- Construct the ChainComplexMap associated to a DGAlgebraMap
    • zerothHomology(DGAlgebra), see zerothHomology -- Compute the zeroth homology of the DGAlgebra A as a ring.
  • Symbols
    • AssertWellDefined -- Option to check whether the lifted map on DGAlgebras is well defined.
    • cycles -- Cycles chosen when computing the homology algebra of a DGAlgebra
    • EndDegree -- Option to specify the degree to stop computing killing cycles and acyclic closure
    • GenDegreeLimit -- Option to specify the maximum degree to look for generators
    • natural -- The underlying algebra of a DGAlgebra.
    • RelDegreeLimit -- Option to specify the maximum degree to look for relations
    • ringMap -- RingMap of a DGAlgebraMap in degree zero.
    • InitializeComplex, see setDiff -- Sets the differential of a DGAlgebra manually.
    • InitializeDegreeZeroHomology, see setDiff -- Sets the differential of a DGAlgebra manually.
    • StartDegree -- Option to specify the degree to start computing the acyclic closure and killing cycles
    • TMOLimit -- Option to specify the maximum arity of the trivial Massey operation

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