Well-definedness, acyclicity, and quasi-isomorphism -- Predicates that check DG-theoretic conditions on DG types

The package exposes three families of predicates for checking conditions that distinguish DG objects from their underlying graded / module-theoretic data.

isWellDefined — verifies that an object satisfies the DG axioms appropriate to its type. For a DGAlgebra or DGModule, the central check is that the stored differential squares to zero on every generator. For a DGSubmodule or DGQuotientModule, the check is d-closure of the defining submodule; for a DGAlgebraMap or DGModuleMap, the check is the chain-map condition d \circ f = f \circ d on every generator.

isAcyclic — returns true when a DG algebra or DG module has H_i = 0 for all i >= 1 up to a finite bound. For objects with infinite hom-degree the user must supply EndDegree.

isQuasiIsomorphism — returns true when a DG algebra map or DG module map induces an isomorphism on H_*. The underlying check is performed on toComplexMap and inherits the Concentration option from the Complexes package.

All three predicates accept the option EndDegree where applicable; for finite-degree objects the default bound is maxDegree of the object and no option is required.

i1 : R = QQ[x, y, z]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o2 : DGAlgebra

i3 : isWellDefined A

o3 = true

i4 : isAcyclic A

o4 = true

i5 : Anat = A.natural

o5 = Anat

o5 : PolynomialRing, 3 skew commutative variable(s)

i6 : M = freeDGModule(A, {0})

o6 = {Base ring => R               }
      DG algebra => Anat
                            1
      Natural module => Anat
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o6 : DGModule

i7 : S = dgSubmodule(M, matrix {{1_Anat}})

o7 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}}
                     1
     natural  => Anat
     inclusion => | 1 |

o7 : DGSubmodule

i8 : Q = M / S

o8 = DGQuotientModule Q = M / S
     Q.natural = cokernel | 1 |
     Degrees   = {{0, 0}}

o8 : DGQuotientModule

i9 : isWellDefined S

o9 = true

i10 : isWellDefined Q

o10 = true

i11 : phi = identityDGAlgebraMap A

o11 = map (Anat, Anat, {T   , T   , T   , x, y, z})
                         1,1   1,2   1,3

o11 : DGAlgebraMap

i12 : isWellDefined phi

o12 = true

i13 : isQuasiIsomorphism phi

o13 = true

For a regular polynomial ring the Koszul DG algebra is acyclic and its identity map is a quasi-isomorphism. Both S and Q = M / S are well-defined because S is d-closed.

Well-definedness, acyclicity, and quasi-isomorphism -- Predicates that check DG-theoretic conditions on DG types

See also

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