Structure of the biserial fiber-kernel module #
The first and last obstructions in the Pogorzały--Skowroński induction are fiber products of two length-three branches over a common simple quotient. This file records the exact length and socle calculations for that construction. The source-specific element calculation used to prove indecomposability is kept separate.
The difference map defining a fiber kernel is surjective as soon as its left branch is surjective.
The finitely generated wrapper of a fiber kernel is linearly equivalent to the literal kernel subtype. Keeping this transport explicit avoids depending on reducibility of the bundled module object.
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The canonical inclusion of the bundled fiber kernel into the ambient binary product.
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The canonical fiber-kernel inclusion is injective.
The product of the two branch radicals inside the ambient product.
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The part of a fiber kernel lying in both branch radicals.
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If both branch maps are onto a nonzero common quotient and the left branch radical is killed by its quotient map, the fiber kernel is not contained in the product of the branch radicals.
Composition length is additive across the short exact sequence defined by a surjective fiber-kernel map.
Two length-three branches over a simple quotient have a fiber kernel of composition length five.
If the simple socles of both branches are killed by the quotient maps, then the socle of their fiber kernel maps onto the product of the two branch socles inside the ambient product.
Under the preceding branch hypotheses, the socle of the fiber kernel is linearly equivalent to the product of the two branch socles.
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If both branch socles are simple, the fiber kernel has socle length two.
Quotient both coordinates of a fiber kernel by their branch socles.
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Quotienting the branch socles maps onto the corresponding fiber kernel of quotient branches.
The kernel of the branch-socle quotient map is precisely the fiber kernel's socle.
Quotienting a fiber kernel by its socle is the fiber kernel of the two branch quotients by their socles.
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For a length-three branch onto a length-one top, the induced map from the quotient by the branch socle kills the simple next socle layer.
If the two branch next socle layers are copies of the same simple module, then the next socle layer of their fiber kernel is their product.
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For two length-three branches over a length-one top, identifying both branch next socles with one simple module computes the next socle of the fiber kernel. The kernel conditions for those branch next socles follow from the length data and surjectivity.