isAcyclic -- Determine whether a DG algebra has no higher homology

Usage:

isAcyc = isAcyclic A
Inputs:
- A, an instance of the type DGAlgebra,
Optional inputs:
- EndDegree => ..., default value -1, Option to specify the degree to finish checking acyclicity
Outputs:
- isAcyc, a Boolean value, true if HH_i A = 0 for all i > 0, and false otherwise

Description

A DG algebra is acyclic (in the convention used by this package) if it has zero homology in all positive degrees, so that A -> HH_0 A is a quasi-isomorphism. isAcyclic checks homology degree by degree up through maxDegree A; for unbounded DG algebras an EndDegree option bounds the search.

The Koszul complex of a regular sequence is acyclic:

i1 : Q = ZZ/101[a,b,c,d]

o1 = Q

o1 : PolynomialRing

i2 : I = ideal(a^4, b^4, c^4, d^4)

             4   4   4   4
o2 = ideal (a , b , c , d )

o2 : Ideal of Q

i3 : isAcyclic(koszulComplexDGA I)

o3 = true

But the Koszul complex of a ring with nontrivial relations in homology is not:

i4 : R = ZZ/101[a,b,c,d] / ideal(a^4 + b^4 + c^4 + d^4)

o4 = R

o4 : QuotientRing

i5 : isAcyclic(koszulComplexDGA R)

o5 = false

An acyclic closure built to sufficient degree is acyclic in the checked range:

i6 : R = ZZ/101[a,b,c,d] / ideal(a^3, b^3, c^3, d^3)

o6 = R

o6 : QuotientRing

i7 : A = acyclicClosure(R, EndDegree => 3)

o7 = {Ring => R                                                   }
      Underlying algebra => R[T   ..T   ]
                               1,1   2,4
                                    2       2       2       2
      Differential => {a, b, c, d, a T   , b T   , c T   , d T   }
                                      1,1     1,2     1,3     1,4

o7 : DGAlgebra

i8 : isAcyclic(A, EndDegree => 3)

o8 = true

Ways to use isAcyclic:

isAcyclic(DGAlgebra)
isAcyclic(DGModule) -- Determine whether a DG module has vanishing positive-degree homology
isAcyclic(DGQuotientModule) -- Determine whether a DG quotient module has vanishing positive-degree homology

For the programmer

The object isAcyclic is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8317:0.

isAcyclic -- Determine whether a DG algebra has no higher homology

Description

See also

Ways to use isAcyclic:

For the programmer