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

## Description

This package is used to define and manipulate DG algebras.

## Version

This documentation describes version 1.1.0 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.
• 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.
• 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
• "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
• "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 .