diff(DGQuotientModule,Vector) -- Apply the induced differential on a DG quotient module to an element

Function: diff
Usage:

dv = diff(Q, v)
Inputs:
- Q, an instance of the type DGQuotientModule, A DG quotient module Q = M / S.
- v, a vector, A homogeneous element of Q.natural.
Outputs:
- dv, a vector, The image d_Q(v) in Q.natural.

Description

Because the inclusion of S into M is a DG submodule (its columns are d-closed in M), the differential of M descends to Q. This method applies the Leibniz rule to a representative in Q.natural without going through a matrix-level differential, which is convenient because Q.natural is a cokernel rather than a free module.

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

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

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

o3 : DGModule

i4 : natGens = apply(rank M.natural, i -> (M.natural)_i)

o4 = {| 1 |, | 0 |}
      | 0 |  | 1 |

o4 : List

i5 : setDiff(M, {0, x * natGens#0})

o5 = {Base ring => R                       }
      DG algebra => R[T   ]
                       1,1
                                 2
      Natural module => (R[T   ])
                            1,1
      Generator degrees => {{0, 0}, {1, 0}}
      Differentials on gens => {0, | x |}
                                   | 0 |

o5 : DGModule

i6 : S = dgSubmodule(M, (id_(M.natural))_{1})

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{1, 0}, {0, 0}}
                          2
     natural  => (R[T   ])
                     1,1
     inclusion => {0, 0} | 0 x |
                  {1, 0} | 1 0 |

o6 : DGSubmodule

i7 : Q = M / S

o7 = DGQuotientModule Q = M / S
     Q.natural = cokernel {0, 0} | 0 x |
                          {1, 0} | 1 0 |
     Degrees   = {{0, 0}, {1, 0}}

o7 : DGQuotientModule

i8 : projection Q

                         2
o8 = {Source => (R[T   ])              }
                    1,1
      Target => cokernel {0, 0} | 0 x |
                         {1, 0} | 1 0 |
      Natural => {0, 0} | 1 0 |
                 {1, 0} | 0 0 |

o8 : DGModuleMap

i9 : diff(Q, (Q.natural)_0)

o9 = 0

o9 : Vector

Ways to use this method:

diff(DGQuotientModule,Vector) -- Apply the induced differential on a DG quotient module to an element

The source of this document is in DGAlgebras/doc.m2:6200:0.

diff(DGQuotientModule,Vector) -- Apply the induced differential on a DG quotient module to an element

Description

See also

Ways to use this method: