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
• acyclicClosure -- Compute the acyclic closure of a DGAlgebra.
• 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(Ring) -- Compute the acyclic closure of the residue field of a ring up to a certain degree
• adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
• AssertWellDefined -- Option to check whether the lifted map on DGAlgebras is well defined.
• blockDiff -- prepares a map for display
• cycles -- Cycles chosen when computing the homology algebra of a DGAlgebra
• deviations -- Computes the deviations of the input ring, complex, or power series.
• deviations(...,DegreeLimit=>...) -- Option to specify the maximum degree to look for generators when computing the deviations
• deviationsToPoincare -- Computes the power series corresponding to a set of deviations.
• DGAlgebra -- The class of all DGAlgebras
• DGAlgebra ** DGAlgebra -- Tensor product of a DGAlgebra and another ring.
• DGAlgebra ** Ring -- Tensor product of a DGAlgebra and another ring.
• DGAlgebraMap -- The class of all DG Algebra maps
• dgAlgebraMap -- Define a DG algebra map between DG algebras.
• 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 -- Shows natural decomposition of a map in the Tate resolution
• EndDegree -- Option to specify the degree to stop computing killing cycles and acyclic closure
• 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=>...) -- Option to specify the maximum degree to look for generators
• findTrivialMasseyOperation(...,TMOLimit=>...) -- Option to specify the maximum arity of a trivial Massey operation, if one exists.
• freeDGAlgebra -- Constructs a DGAlgebra
• GenDegreeLimit -- Option to specify the maximum degree to look for generators
• getBasis -- Get a basis for a particular homological degree of a DG algebra.
• getBasis(...,Limit=>...) -- Option to specify the maximum number of basis elements to return
• 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 -- Compute a presentation of M_i as an R-module
• getGenerators -- Returns a list of cycles whose images generate HH(A) as an algebra
• getGenerators(...,DegreeLimit=>...) -- Option to specify the degree to stop finding generators of HH(DGAlgebra)
• getGenerators(...,StartDegree=>...) -- Option to specify the degree to start finding generators of HH(DGAlgebra)
• getGenerators(...,Verbosity=>...) -- Option to specify the maximum degree to look for generators when computing the deviations
• 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 -- Compute the homology algebra of a DGAlgebra.
• homologyAlgebra(...,GenDegreeLimit=>...) -- Option to specify the maximum degree to look for generators
• homologyAlgebra(...,RelDegreeLimit=>...) -- Option to specify the maximum degree to look for relations
• homologyAlgebra(...,Verbosity=>...) -- Option to specify the maximum degree to look for generators when computing the deviations
• 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.
• 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=>...) -- Option to specify the maximum degree to look for generators
• isGolodHomomorphism(...,TMOLimit=>...) -- Option to specify the maximum degree to look for generators when computing the deviations
• 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=>...) -- Option to specify the maximum degree to look for generators
• killCycles -- Adjoins variables to make non-bounding cycles boundaries in the lowest positive degree with nontrivial homology.
• killCycles(...,EndDegree=>...) -- Option to specify the degree to stop looking for cycles
• killCycles(...,StartDegree=>...) -- Option to specify the degree to start looking for cycles
• 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 -- 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
• masseyTripleProduct(DGAlgebra,ZZ,ZZ,ZZ) -- Computes the matrix representing all triple Massey operations.
• maxDegree -- Computes the maximum homological degree of a DGAlgebra
• natural -- The underlying algebra of a DGAlgebra.
• net(DGAlgebra) -- Outputs the pertinent information about a DGAlgebra
• net(DGAlgebraMap) -- Outputs the pertinent information about a DGAlgebraMap
• RelDegreeLimit -- Option to specify the maximum degree to look for relations
• setDiff -- Sets the differential of a DGAlgebra manually.
• source(DGAlgebraMap) -- Outputs the source of a DGAlgebraMap
• StartDegree -- Option to specify the degree to start computing the acyclic closure and killing cycles
• target(DGAlgebraMap) -- Outputs the target of a DGAlgebraMap
• TMOLimit -- Option to specify the maximum arity of the trivial Massey operation
• toComplex -- Converts a DGAlgebra to a ChainComplex
• toComplex(DGAlgebra,ZZ) -- Converts a DGAlgebra to a ChainComplex
• toComplexMap -- Construct the ChainComplexMap associated to a DGAlgebraMap
• torAlgebra -- Computes the Tor algebra of a ring
• torAlgebra(...,GenDegreeLimit=>...) -- Option to specify the maximum degree to look for generators
• torAlgebra(...,RelDegreeLimit=>...) -- Option to specify the maximum degree to look for 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.
• zerothHomology -- Compute the zeroth homology of the DGAlgebra A as a ring.