DGAlgebraMap -- The class of DG algebra maps

Description

A DGAlgebraMap represents a morphism of DG algebras. It carries four pieces of data: a source and target (each a DGAlgebra), an underlying ring map f.natural on the graded-commutative algebras, and a degree-zero ring map f.ringMap between the underlying rings. A DG algebra map is a ring map that commutes with the differentials, so that d_B \circ f = f \circ d_A where d_A and d_B are the differentials of the source and target.

There are two principal ways to construct a DG algebra map: dgAlgebraMap builds one from a matrix specifying where the DG generators go, and liftToDGMap lifts a ring map on the degree-zero part to a morphism of acyclic closures or Koszul complexes.

i1 : R = ZZ/101[a,b,c]/ideal{a^3, b^3, c^3}

o1 = R

o1 : QuotientRing

i2 : S = R / ideal{a^2*b^2*c^2}

o2 = S

o2 : QuotientRing

i3 : A = acyclicClosure(R, EndDegree => 3)

o3 = {Ring => R                                        }
      Underlying algebra => R[T   ..T   ]
                               1,1   2,3
                                 2       2       2
      Differential => {a, b, c, a T   , b T   , c T   }
                                   1,1     1,2     1,3

o3 : DGAlgebra

i4 : B = acyclicClosure(S, EndDegree => 3)

o4 = {Ring => S                                                                                                                                                                    }
      Underlying algebra => S[T   ..T   , T   ..T   , T   ..T   , T   ..T   ]
                               1,1   1,3   2,1   2,4   3,1   3,3   4,1   4,6
                                 2       2       2         2 2       2 2        2 2        2 2       2 2            2 2           2 2            2           2           2
      Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   , b c T   T   , -a c T   T   , b c T   T   , -a T   T   , c T   T   , b T   T   }
                                   1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4

o4 : DGAlgebra

i5 : phi = liftToDGMap(B, A, map(S, R))

o5 = map (S[T   ..T   , T   ..T   , T   ..T   , T   ..T   ], R[T   ..T   ], {T   , T   , T   , T   , T   , T   , a, b, c})
             1,1   1,3   2,1   2,4   3,1   3,3   4,1   4,6      1,1   2,3     1,1   1,2   1,3   2,1   2,2   2,3

o5 : DGAlgebraMap

i6 : source phi === A

o6 = true

i7 : target phi === B

o7 = true

Once constructed, a DG algebra map can be checked with isWellDefined(DGAlgebraMap), converted to a ComplexMap via toComplexMap, or pushed through homology with HH DGAlgebraMap.

Functions and methods returning an object of class DGAlgebraMap:

dgAlgebraMap -- Define a DG algebra map between DG algebras.
identityDGAlgebraMap -- The identity DGAlgebraMap on a DG algebra
liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism

Methods that use an object of class DGAlgebraMap:

isWellDefined(DGAlgebraMap) -- see dgAlgebraMap -- Define a DG algebra map between DG algebras.
DGAlgebraMap * DGAlgebraMap -- Composition of DG algebra maps
DGAlgebraMap ** DGAlgebraMap -- Tensor product of DG algebra maps
DGAlgebraMap ** Ring -- Base change of a DG algebra map along a ring map
DGAlgebraMap == DGAlgebraMap (missing documentation)
HH DGAlgebraMap -- The induced ring map on homology algebras
isQuasiIsomorphism(DGAlgebraMap) -- Determine whether a DG algebra map induces an isomorphism on homology
net(DGAlgebraMap) -- Short-form display of a DG algebra map
minimalPresentation(DGAlgebraMap) -- see prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
prune(DGAlgebraMap) -- Pruning a DG algebra map is the identity
source(DGAlgebraMap) -- The source of a DG algebra map
target(DGAlgebraMap) -- The target of a DG algebra map
toComplexMap(DGAlgebraMap) -- see toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap
toComplexMap(DGAlgebraMap,ZZ) -- see toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap

For the programmer

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8959:0.

DGAlgebraMap -- The class of DG algebra maps

Description

See also

Functions and methods returning an object of class DGAlgebraMap:

Methods that use an object of class DGAlgebraMap:

For the programmer