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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.

Author

Version

This documentation describes version 1.1.1--with fix of killCycles and new code and docs for displayBlockDiff and blockDiff 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 the acyclic closure of a DGAlgebra.
    • adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
    • 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.
    • dgAlgebraMap -- Define a DG algebra map between DG algebras.
    • dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
    • displayBlockDiff -- Shows natural decomposition of a map in the Tate resolution
    • 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.
    • getBoundaryPreimage -- Attempt to find a preimage of a boundary under the differential of a DGAlgebra.
    • 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.
    • 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.
    • 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
    • masseyTripleProduct -- Computes the Massey triple product of a set of cycles or homology classes
    • 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 the acyclic closure of a DGAlgebra.
    • acyclicClosure(Ring) -- Compute the acyclic closure of the residue field of a ring up to a certain degree
    • adjoinVariables(DGAlgebra,List) -- see adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
    • blockDiff(DGAlgebra,ZZ) -- see blockDiff -- prepares a map for display
    • deviations(ChainComplex) -- 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 -- 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.
    • dgAlgebraMultMap(DGAlgebra,RingElement) -- see dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.
    • diff(DGAlgebra,RingElement) -- Computes the differential of a ring element in a DGAlgebra
    • 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
    • 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 -- Constructs a DGAlgebra
    • getBasis(ZZ,DGAlgebra) -- see getBasis -- Get a basis for a particular homological degree of a DG algebra.
    • 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.
    • getDegNModule(ZZ,Ring,Ring) -- see getDegNModule -- Compute a presentation of M_i as an R-module
    • 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.
    • HH_ZZ DGAlgebra -- Computes the homology of a DG algebra as a module
    • 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.
    • homologyModule(DGAlgebra,Module) -- see homologyModule -- Compute the homology of a DGModule as a module over a DGAlgebra.
    • isAcyclic(DGAlgebra) -- see isAcyclic -- Determines if a DGAlgebra is acyclic.
    • 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.
    • 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.
    • koszulComplexDGA(Ring) -- see koszulComplexDGA -- Returns the Koszul complex as a DGAlgebra
    • koszulComplexDGA(Ideal) -- Returns the Koszul complex as a DGAlgebra
    • koszulComplexDGA(List) -- Define the Koszul complex on a list of elements as a DGAlgebra
    • liftToDGMap(DGAlgebra,DGAlgebra,RingMap) -- see liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
    • 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
    • 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
    • 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.
    • 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
    • ringMap -- see DGAlgebraMap -- The class of all DG Algebra maps
    • 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
    • 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

For the programmer

The object DGAlgebras is a package.

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