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DGModuleMap == DGModuleMap -- Elementary predicates and constructors for DGModuleMaps

Description

Auxiliary operations mirroring those on ComplexMap. For maps f, g between DG modules over a common DG algebra:

f == g -- equality of source, target, and underlying natural matrix.

isHomogeneous f -- agrees with isHomogeneous f.natural.

map(N, M, 0) -- returns the zero map M -> N. map(M, M, 1) -- returns the identity on M.

isQuasiIsomorphism f -- returns true iff the induced chain map on toComplexMap is a quasi-isomorphism.

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

o1 = R

o1 : QuotientRing
 
i2 : Mdg = minimalSemifreeResolution(koszulComplexDGA R, R^1 / ideal(x,y), EndDegree => 2)

o2 = {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    |

o2 : DGModule
 
i3 : idM = identityDGModuleMap Mdg

                               3
o3 = {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 |

o3 : DGModuleMap
 
i4 : zM = zeroDGModuleMap(Mdg, Mdg)

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

o4 : DGModuleMap
 
i5 : idM == idM

o5 = true
 
i6 : map(Mdg, Mdg, 0) == zM

o6 = true
 
i7 : map(Mdg, Mdg, 1) == idM

o7 = true
 
i8 : isQuasiIsomorphism idM

o8 = true

See also

Ways to use this method:

The source of this document is in DGAlgebras/doc.m2:553:0.