Building DG modules, submodules, and quotients -- The DGModule / DGSubmodule / DGQuotientModule hierarchy and its constructors

The package provides three related types of DG-module-like objects over a fixed DGAlgebra A:

DGModule — a free graded A.natural-module equipped with a differential that satisfies the Leibniz rule against A. Built via freeDGModule, with differentials set by setDiff.

DGSubmodule — a subobject of a DGModule M, represented by a matrix of generators whose column span is closed under the differential of M. Built via dgSubmodule, which d-saturates the seed generators automatically.

DGQuotientModule — the cokernel M / S of the inclusion S → M, with the induced differential. Built via dgQuotientModule or the shorthand M / S.

Every object in this hierarchy carries an underlying graded A.natural-module accessible via M.natural, a list of multi-degrees M.Degrees, and a list of differentials M.diff (one per natural generator for DGModule and DGSubmodule; per quotient-presentation generator for DGQuotientModule). Structural correctness can be checked with isWellDefined.

i1 : R = ZZ/101[x, y]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : M = freeDGModule(A, {0, 1})

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

o3 : DGModule

i4 : isWellDefined M

o4 = true

i5 : Anat = A.natural

o5 = Anat

o5 : PolynomialRing, 2 skew commutative variable(s)

i6 : S = dgSubmodule(M, (id_(M.natural))_{0})

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}}
                     1
     natural  => Anat
     inclusion => {0, 0} | 1 |
                  {1, 0} | 0 |

o6 : DGSubmodule

i7 : isWellDefined S

o7 = true

i8 : Q = M / S

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

o8 : DGQuotientModule

i9 : isWellDefined Q

o9 = true

Building DG modules, submodules, and quotients -- The DGModule / DGSubmodule / DGQuotientModule hierarchy and its constructors

See also

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