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

Function: isAcyclic
Usage:

b = isAcyclic M

b = isAcyclic(M, EndDegree => n)
Inputs:
- M, an instance of the type DGModule,
Optional inputs:
- EndDegree => an integer, default value -1, An upper bound for the range of degrees checked. Required when M has infinite maxDegree (for example, when its DG algebra is an acyclicClosure). When omitted and M is finite-dimensional, maxDegree M is used.
Outputs:
- b, a Boolean value, True when H_i(M) = 0 for all 1 <= i <= n.

Description

Checks per-degree homology via HH_ZZ DGModule and prune. Degree zero is intentionally excluded: acyclicity concerns only positive-degree homology.

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 : KM = koszulComplexDGM R^1

o3 = {Base ring => R                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,3
                                       1
      Natural module => (R[T   ..T   ])
                            1,1   1,3
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o3 : DGModule

i4 : isAcyclic KM

o4 = true

For the Koszul DG module on a regular sequence, all positive-degree homology vanishes (H_0 = R / (x, y, z) is the only nonzero piece).

i5 : R' = QQ[x, y]/ideal(x^2, y^2)

o5 = R'

o5 : QuotientRing

i6 : KM' = koszulComplexDGM R'^1

o6 = {Base ring => R'                    }
      DG algebra => R'[T   ..T   ]
                        1,1   1,2
                                        1
      Natural module => (R'[T   ..T   ])
                             1,1   1,2
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o6 : DGModule

i7 : not isAcyclic(KM', EndDegree => 3)

o7 = true

Over a non-regular ring the Koszul DG module picks up nontrivial positive-degree homology.

Ways to use this method:

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

The source of this document is in DGAlgebras/doc.m2:5163:0.

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

Description

See also

Ways to use this method: