kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source

Function: kernel
Usage:

S = kernel f
Inputs:
- f, an instance of the type DGModuleMap, A morphism M -> N of DG modules over a common DG algebra.
Optional inputs:
- DegreeLimit (missing documentation) => ..., default value {},
- Strategy (missing documentation) => ..., default value {},
- SubringLimit => ..., default value infinity,
Outputs:
- S, an instance of the type DGSubmodule, The DG submodule of M = source f generated by the kernel of f.natural as a module over M.dgAlgebra.natural.

Description

A chain map sends cycles to cycles, so the kernel of f.natural at the module level is already d-closed. The constructor takes the natural-level kernel generators and wraps them with dgSubmodule.

Mathematically interesting example: over R = k[x, y]/(x^2, y^2) the endomorphism "multiplication by x" of the Koszul DG module KM squares to zero, so its kernel contains its image:

i1 : R = ZZ/101[x, y] / ideal(x^2, y^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : KM = koszulComplexDGM R^1

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

o3 : DGModule

i4 : mx = dgModuleMap(KM, KM, x * id_(KM.natural))

                               1
o4 = {Source => (R[T   ..T   ]) }
                    1,1   1,2
                               1
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => | x |

o4 : DGModuleMap

i5 : kerMx = kernel mx

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}}
                                1
     natural  => (R[T   ..T   ])
                     1,1   1,2
     inclusion => | x |

o5 : DGSubmodule

i6 : imgMx = image mx

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

o6 : DGSubmodule

i7 : isWellDefined kerMx

o7 = true

i8 : numcols (inclusion kerMx).natural

o8 = 1

i9 : isSubset(imgMx, kerMx)

o9 = true

The inclusion of a DG submodule is injective, so its kernel is zero:

i10 : Sfull = dgSubmodule(KM, id_(KM.natural));

i11 : isZero kernel (inclusion Sfull)

o11 = true

Ways to use this method:

kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4063:0.

kernel(DGModuleMap) -- Kernel of a DG module map as a DG submodule of its source

Description

See also

Ways to use this method: