dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module

Usage:

t = dgModuleSummary M

t = dgModuleSummary(M, n)
Inputs:
- M, an instance of the type DGModule, A free DG module (for example from freeDGModule or minimalSemifreeResolution).
- n, an integer, Upper bound on the hom-degree range to display. When omitted, maxDegree M is used; if that is infinity the method raises an error asking for an explicit bound.
Outputs:
- t, a net, A centered table with one row per hom-degree in 0..n. Columns are the hom-degree k, the number of generators of M placed at hom-degree k (i.e. adjoined in that degree), and the rank of F_k as an A.ring-module.

Description

A one-call overview of how the generators of M distribute across hom-degrees and how that translates into base-ring ranks. Matches the data fed into moduleDifferential.

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

o1 = R

o1 : QuotientRing

i2 : k = R^1 / ideal(x, y)

o2 = cokernel | x y |

                            1
o2 : R-module, quotient of R

i3 : A = koszulComplexDGA R

o3 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o3 : DGAlgebra

i4 : Mdg = minimalSemifreeResolution(A, k, EndDegree => 3)

o4 = {Base ring => R                                                                                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       6
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}, {4, 4}, {4, 4}, {4, 4}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |, |     0    |, |     0    |, |     0    |}
                                   |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |  |     0    |
                                   |     0    |  |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |

o4 : DGModule

i5 : instance(dgModuleSummary(Mdg, 3), Net)

o5 = true

Caveat

The one-argument form is only applicable when the underlying hom-grading of M is bounded (for example when M is a koszulComplexDGM output over a polynomial ring). For arbitrary semifree DG modules, pass an explicit upper bound.

Ways to use dgModuleSummary:

dgModuleSummary(DGModule)
dgModuleSummary(DGModule,ZZ)

For the programmer

The object dgModuleSummary is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:6289:0.

dgModuleSummary -- Tabulate hom-degree-wise generator counts and free-rank counts for a DG module

Description

Caveat

See also

Ways to use dgModuleSummary:

For the programmer