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# DGAlgebra -- The class of all DGAlgebras

## Description

Some common ways to create DGAlgebras include koszulComplexDGA, freeDGAlgebra, setDiff, and acyclicClosure.

## Functions and methods returning an object of class DGAlgebra :

• acyclicClosure -- Compute the acyclic closure of a DGAlgebra.
• adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
• freeDGAlgebra -- Constructs a DGAlgebra
• 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
• setDiff -- Sets the differential of a DGAlgebra manually.

## Methods that use an object of class DGAlgebra :

• acyclicClosure(DGAlgebra) -- see acyclicClosure -- Compute the acyclic closure of a DGAlgebra.
• adjoinVariables(DGAlgebra,List) -- see adjoinVariables -- Adjoins variables to make the specified cycles boundaries.
• blockDiff(DGAlgebra,ZZ) -- see blockDiff -- prepares a map for display
• 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.
• 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
• 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.
• 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.
• 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_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.
• 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
• 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
• setDiff(DGAlgebra,List) -- see setDiff -- Sets the differential of a DGAlgebra manually.
• toComplex(DGAlgebra) -- see toComplex -- Converts a DGAlgebra to a ChainComplex
• toComplex(DGAlgebra,ZZ) -- Converts a DGAlgebra to a ChainComplex
• zerothHomology(DGAlgebra) -- see zerothHomology -- Compute the zeroth homology of the DGAlgebra A as a ring.

## For the programmer

The object DGAlgebra is a type, with ancestor classes MutableHashTable < HashTable < Thing.