moduleDifferential -- The hom-degree-n differential of a DG module as a matrix over the base ring

Usage:

dn = moduleDifferential(n, M)
Inputs:
- n, an integer, A hom-degree.
- M, an instance of the type DGModule, A DG module over a DG algebra A; must be free (as in freeDGModule output) or in koszulComplexDGM form.
Outputs:
- dn, a matrix, The component d_n : F_n -> F_{n-1} of the differential of M, as a homogeneous matrix over A.ring. Columns are indexed by the pairs (i, m) of moduleBasisPairs(M, n); rows by the pairs of moduleBasisPairs(M, n-1).

Description

Two code paths are used internally:

When M comes from koszulComplexDGM (so M.module is set) and every generator differential is zero, d_n is the tensor product polyDifferential(n, A) \otimes id_{M.module}. This is the fast path.

For general free DG modules, d_n is computed monomial-by-monomial via the Leibniz rule, extracting A.ring-linear coefficients in the target basis. Results are cached per hom-degree in M.cache.diffs.

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 => 2)

o4 = {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    |

o4 : DGModule

i5 : d1 = moduleDifferential(1, Mdg)

o5 = | x y |

             1      2
o5 : Matrix R  <-- R

i6 : d2 = moduleDifferential(2, Mdg)

o6 = {1} | -y x 0 |
     {1} | x  0 y |

             2      3
o6 : Matrix R  <-- R

i7 : d1 * d2 == 0

o7 = true

Every column of d_{n-1} d_n is zero, which is the d^2 = 0 condition.

Caveat

Raises an error on non-free DG modules (for example, the natural module of a DGQuotientModule). In that setting, compute the differential at the element level via diff(Q, v) or work with dgComplex.

Ways to use moduleDifferential:

moduleDifferential(ZZ,DGModule)

For the programmer

The object moduleDifferential is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:6111:0.

moduleDifferential -- The hom-degree-n differential of a DG module as a matrix over the base ring

Description

Caveat

See also

Ways to use moduleDifferential:

For the programmer