oalg-abg-2.0.0.0: Finitely generated abelian groups.

Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	None
Language	Haskell2010

OAlg.AbelianGroup.KernelsAndCokernels

Contents

Kernels
Cokernels
Smith Normal
Adjunction

Description

Kernels and Cokernels for homomorphisms between finitely generated abelian groups.

Synopsis

abhKernels :: Kernels N1 AbHom
abhKernelsFreeFromG :: forall (k :: N'). Attestable k => KernelsG (ConicFreeTip Cone) (SliceDiagram (Free k)) N1 AbHom
abhKernelsFreeFromG' :: forall (k :: N') q. Attestable k => q k -> KernelsG (ConicFreeTip Cone) (SliceDiagram (Free k)) N1 AbHom
abhKernelsSomeFreeFreeTip :: KernelsSomeFreeFreeTip AbHom
abhCokernels :: Cokernels N1 AbHom
abhCokernelsLftFreeG :: CokernelsG ConeLiftable Diagram N1 AbHom
abhCokernelsLiftableSomeFree :: CokernelsG ConeLiftable SomeFreeSliceDiagram N1 AbHom
isoSmithNormal :: AbGroup -> Inv AbHom
abhSliceFreeAdjunction :: forall (k :: N'). Attestable k => Any k -> Adjunction (SliceAdjunction (Free k) Cone AbHom) (SliceFactor 'From (Free k) AbHom) (SliceFactor 'To (Free k) AbHom)

Kernels

abhKernels :: Kernels N1 AbHom Source #

kernels for AbHom.

abhKernelsFreeFromG :: forall (k :: N'). Attestable k => KernelsG (ConicFreeTip Cone) (SliceDiagram (Free k)) N1 AbHom Source #

kernels with a free tip of free sliced diagrams.

abhKernelsFreeFromG' :: forall (k :: N') q. Attestable k => q k -> KernelsG (ConicFreeTip Cone) (SliceDiagram (Free k)) N1 AbHom Source #

kernels with a free tip of free sliced diagrams according to the given proxy type.

abhKernelsSomeFreeFreeTip :: KernelsSomeFreeFreeTip AbHom Source #

kernels with a free tip for free sliced From diagrams.

Cokernels

abhCokernels :: Cokernels N1 AbHom Source #

cokernels.

abhCokernelsLftFreeG :: CokernelsG ConeLiftable Diagram N1 AbHom Source #

liftable cokernels.

abhCokernelsLiftableSomeFree :: CokernelsG ConeLiftable SomeFreeSliceDiagram N1 AbHom Source #

liftable cokernels for free sliced To diagrams.

Smith Normal

isoSmithNormal :: AbGroup -> Inv AbHom Source #

isomorphism to its smith normal group.

Properties Let g be in AbGroup, then holds:

start (isoSmithNormal g) == g.
end (isoSmithNormal g) is smith normal (see definition AbGroup).

Adjunction

abhSliceFreeAdjunction :: forall (k :: N'). Attestable k => Any k -> Adjunction (SliceAdjunction (Free k) Cone AbHom) (SliceFactor 'From (Free k) AbHom) (SliceFactor 'To (Free k) AbHom) Source #

the cokernel-kernel adjunction for a given Free k.