Cokernel subquotient obstructions for biseriality #
This is the cokernel counterpart of BiserialKernelObstruction. If a map
into a binary product lands inside two chosen branch submodules, the cokernel
still surjects onto the product of the two branch quotients. Equal nonzero
branch quotients therefore give the repeated self-subquotient used in the
cokernel contradictions of the Pogorzały--Skowroński induction.
The finitely generated cokernel of a linear map.
Instances For
If a map into Y × Z lands in QY × QZ, its cokernel surjects
onto (Y/QY) × (Z/QZ). When both quotients are the same nonzero module,
this is a repeated self-subquotient certificate.
Complete coordinate thinness rules out indecomposability of the cokernel configuration above. This is the contradiction endpoint for the cokernel modules in the direct biserial induction.
An injective copy of a product of two branch submodules, each with the same nonzero quotient, gives a repeated self-subquotient of the ambient module.
If a product of two branch submodules meets the image of a map trivially, that branch product embeds in the cokernel. Equal nonzero branch quotients then give a repeated self-subquotient of the cokernel.
Complete coordinate thinness rules out indecomposability of the disjoint-branch cokernel configuration. This is the direct endpoint for the diagonal-cokernel obstruction in the biserial induction.
If a diagonal copy of a submodule is killed inside two injective ambient branches, the resulting cokernel contains a submodule surjecting onto two copies of the common quotient.
A surjection from a local module to a simple module identifies the simple target with the source top.
Instances For
A map between local modules vanishes when the target radical is simple and the source top is nonisomorphic to both simple layers of the target. Indeed, a nonzero image is either the whole target or its simple radical; either case makes one of those layers a simple quotient of the source.
A linear map between nonisomorphic simple modules is zero.
A map to a local module with simple radical vanishes when the source radical is the sum of two simple modules, neither of which is isomorphic to the target radical, and the source top is also nonisomorphic to that radical. A surjective map would have to map the source radical onto the target radical, while a proper nonzero image would itself be the target radical.
The cokernel obtained by gluing two indecomposable branches along a nonzero diagonal submodule is indecomposable if there are no cross maps from either branch to the quotient of the other by the glued image.
The proof tests an idempotent endomorphism of the cokernel. Cross-map vanishing forces it to preserve both canonical branch images. Its two restrictions are therefore zero or one by branch indecomposability, and the nonzero intersection of the branch images forces those choices to agree.
A source-facing form of the diagonal-cokernel criterion. It is enough that both branches have simple top and that each quotient by the glued image has simple top and simple radical, with neither layer isomorphic to the opposite branch top.
Complete coordinate thinness therefore forces every such diagonal submodule cokernel to be decomposable.