degrees(DGModule) -- Multi-degrees of the natural generators of a DG module

Function: degrees
Usage:

ds = degrees M
Inputs:
- M, an instance of the type DGModule, A DGModule, DGSubmodule, or DGQuotientModule.
Outputs:
- ds, a list, A list of multi-degrees, one per natural generator. Each multi-degree is a list whose first entry is the homological degree and whose remaining entries are the internal degrees.

Description

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, 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}, {1, 0}, {2, 0}}
      Differentials on gens => {0, 0, 0}

o3 : DGModule

i4 : degrees M == {{0,0},{1,0},{2,0}}

o4 = true

Internal degrees come from the ambient ring's grading and are concatenated onto the homological degree.

i5 : Rg = ZZ/101[x, y, Degrees => {{1,0}, {0,1}}]

o5 = Rg

o5 : PolynomialRing

i6 : Ag = koszulComplexDGA Rg

o6 = {Ring => Rg                          }
      Underlying algebra => Rg[T   ..T   ]
                                1,1   1,2
      Differential => {x, y}

o6 : DGAlgebra

i7 : Mg = freeDGModule(Ag, {{0,0,0}, {1,0,0}})

o7 = {Base ring => Rg                            }
      DG algebra => Rg[T   ..T   ]
                        1,1   1,2
                                        2
      Natural module => (Rg[T   ..T   ])
                             1,1   1,2
      Generator degrees => {{0, 0, 0}, {1, 0, 0}}
      Differentials on gens => {0, 0}

o7 : DGModule

i8 : degrees Mg

o8 = {{0, 0, 0}, {1, 0, 0}}

o8 : List

Ways to use this method:

degrees(DGModule) -- Multi-degrees of the natural generators of a DG module
degrees(DGQuotientModule)
degrees(DGSubmodule)

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:3536:0.

degrees(DGModule) -- Multi-degrees of the natural generators of a DG module

Description

See also

Ways to use this method: