isWellDefined(DGModuleMap) -- Check that a DG module map has correct structure and is a chain map

Function: isWellDefined
Usage:

b = isWellDefined f
Inputs:
- f, an instance of the type DGModuleMap,
Outputs:
- b, a Boolean value, True when f has all expected structural keys, source and target are DG modules (or DG quotient modules) over a common DG algebra, the natural matrix has the correct source and target, f preserves hom-degree, and d_N(f(e_i)) = f(d_M(e_i)) on every natural generator e_i of the source.

Description

Six checks in order: structural key shape; source and target types are DGModule or DGQuotientModule; source and target share the same DG algebra; f.natural is a module map with the correct source and target; the hom-degree of f is zero; and the chain-map condition holds on every generator of the source.

Modeled on isWellDefined ComplexMap from the Complexes package. Diagnostic messages are emitted when debugLevel > 0.

i1 : R = QQ[x, y]/ideal(x^2, y^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o2 : DGAlgebra

i3 : M = minimalSemifreeResolution(A, R^1 / ideal(x, y), EndDegree => 2)

o3 = {Base ring => R                                          }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       3
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
                                   |     0    |  |     0    |
                                   |     0    |  |     0    |

o3 : DGModule

i4 : idM = identityDGModuleMap M

                               3
o4 = {Source => (R[T   ..T   ])  }
                    1,1   1,2
                               3
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => {0, 0} | 1 0 0 |
                 {2, 2} | 0 1 0 |
                 {2, 2} | 0 0 1 |

o4 : DGModuleMap

i5 : zM = zeroDGModuleMap(M, M)

                               3
o5 = {Source => (R[T   ..T   ]) }
                    1,1   1,2
                               3
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => 0

o5 : DGModuleMap

i6 : isWellDefined idM

o6 = true

i7 : isWellDefined zM

o7 = true

Ways to use this method:

isWellDefined(DGModuleMap) -- Check that a DG module map has correct structure and is a chain map

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:5115:0.

isWellDefined(DGModuleMap) -- Check that a DG module map has correct structure and is a chain map

Description

See also

Ways to use this method: