Direct biserial induction from coordinate thinness #
This file follows only the biserial part of Pogorzały--Skowroński, Proposition 1. Representation-finiteness and the Schurian conclusion of that proposition are not part of the magnitude proof and are not formalized here.
Induction statement for local modules strictly below a composition length bound.
Instances For
A local biserial induction hypothesis makes a smaller local module uniserial as soon as its radical is again local.
For a nonsimple local module, quotienting by the nonzero socle gives a strictly smaller local module, so the local biserial induction hypothesis applies to it.
A coordinate-thin biserial local module with two disjoint simple submodules becomes length three when quotienting by either simple makes it uniserial. The two simples are forced to be the two radical branches.
Under coordinate thinness, the two complementary simple summands in the top of the radical of a nonuniserial local biserial module are nonisomorphic. Otherwise that radical layer itself is a repeated self-subquotient of the original module.
A finitely generated module with simple socle is one of the indecomposables controlled by the all-coordinate-thin premise.
Under a complete coordinate family, a simple-socle module cannot contain a repeated self-subquotient. This is the dual induction endpoint used in the indecomposability arguments of the direct biserial proof.
A fiber kernel with two nonisomorphic simple coordinate socles is indecomposable if every submodule escaping the product of the branch radicals contains a vector with two nonzero socle coordinates. In a hypothetical direct sum, the two coordinate socles must split between the summands, while the summand carrying the common top contains such a mixed vector.
If the ambient next socle layer is two copies of one simple module, a complement with no repeated self-subquotient can contribute at most one copy. Indeed, a length-two contribution leaves the other complement with zero next socle, so the product decomposition identifies that contribution with the whole homogeneous ambient layer.
The source-shaped indecomposability package for the fiber kernels in
the direct proof. The standard branch-radical preimage is proper, the two
successive socle layers force hypothetical complements to be uniserial, and
hcomplementCapture is precisely the paper's calculation for the uniserial
complement escaping the ambient radical.
Complete coordinate thinness contradicts a fiber kernel satisfying the source's two-socle-layer, element-capture, and equal-branch data. This assembles the first and last kernel contradictions of Proposition 1.
The first source obstruction with its literal quotient branches. Two
disjoint simple layers S,T ⊆ X give Y=X/S and Z=X/T; their common
next socle is constructed through the third isomorphism theorem, and their
maps to the simple top of X are the canonical nested-quotient maps. The
second radical layer S ⊕ T and vanishing third radical layer now drive
the source's element calculation internally, so no complement-capture
hypothesis remains.
In the radical-square configuration of the first source obstruction, quotienting by either simple branch produces a uniserial length-three module. The other branch is the simple radical of the branch radical.
The first source obstruction starting from the literal module L.
Here X is definitionally L / rad³ L; its simple top and vanishing third
radical layer are therefore conclusions of the truncation construction, not
premises supplied by the caller.
Complementary simple branches of rad² L / rad³ L embed as the
two exact second-radical branches of X = L / rad³ L. If the intervening
layer rad L / rad² L is simple, the first fiber-kernel obstruction applies.
A biserial but nonuniserial radical cannot itself have simple top under the coordinate-thin hypothesis.
In the direct induction, a nonuniserial local module cannot have a local radical: the induction hypothesis makes that radical biserial, and the first Pogorzały--Skowroński obstruction gives the contradiction.
If rad² L is nonzero, the smaller local quotient L / rad² L
allows the induction hypothesis to split rad L / rad² L into two
nonisomorphic simple summands.
In the nonzero radical-square branch, the two simple factors of
rad L / rad² L lift to proper nonisomorphic local submodules which span
rad L.
If the two local submodules spanning rad L are disjoint, the smaller
quotients by the opposite branches make both submodules uniserial.
If a local branch M is proper and not contained in the other branch
N, induction and biseriality bound a semisimple intersection by length
two.
The two local branches spanning rad L are incomparable. Otherwise
one branch would be the whole radical, contrary to the already established
nonlocality of rad L.
The intersection of the two nonisomorphic local submodules spanning the
radical is semisimple. A hypothetical nonsemisimple intersection supplies
a maximal nonsimple uniserial submodule. Biseriality of the two smaller
sides and of L / soc L produces the ambient-or-quotient carriers, whose
diagonal cokernel gives the coordinate-thin contradiction.
Once the nonzero intersection in the source proof is known to be semisimple, it has composition length at most two.
If the semisimple branch intersection has composition length two, its images inside both local branches are exactly their socles.
A semisimple length-two intersection splits into complementary nonisomorphic simples. Isomorphic summands would themselves be a repeated self-subquotient of the indecomposable local ambient module.
In the nonzero-intersection branch, semisimplicity and exclusion of composition length two reduce the intersection to a simple module.
Every summand of an intersection identified with the left branch socle maps into the left branch radical, provided the branches are incomparable.
Right-hand version of
infSummand_map_left_le_jacobson_of_eq_mapped_socle.
The semisimple length-two intersection branch is impossible. Gluing one simple intersection summand produces a cokernel which the ambient coordinate argument makes indecomposable, while its repeated complementary summand forbids indecomposability under the global thinness hypothesis.
Once the nonzero branch intersection is semisimple, it is simple. The only other possible positive length was two, and the diagonal-cokernel obstruction excludes that case.
The nonzero intersection of the two local branches supplied by the radical-top construction is simple. Semisimplicity and the exclusion of length two are both consequences rather than inputs.
A nonzero radical submodule gives a smaller local quotient. If induction makes that quotient biserial, coordinate-thin branch placement makes the image of any simple-top radical submodule uniserial.
After the branch intersection has been proved simple, induction on the quotient by that intersection makes both branch quotients uniserial.
If one branch is still nonuniserial after its quotient by the simple intersection has become uniserial, it contains a second simple submodule disjoint from the intersection, as in the final source obstruction.
The two uniserial branch quotients attached to a simple intersection and an auxiliary disjoint simple force the nonuniserial branch to have length three.
Quotienting by the auxiliary simple submodule on one branch embeds the other branch into a smaller biserial local module, so coordinate-thin branch placement makes that other branch uniserial.
The two quotients in the final source fiber kernel have nonisomorphic
simple socles. The long quotient L/S has two uniserial radical branches
with the same simple socle, while L/N is itself uniserial.
The element calculation in the final, asymmetric source fiber kernel. The square of the ring radical sends a lift of the common top to one nonzero vector in each simple summand of the short branch radical. Its action on the short branch itself vanishes, so the discrepancy between the two lifts is killed after quotienting by the long branch.
The final asymmetric fiber kernel is simultaneously indecomposable and forbidden by coordinate thinness. Indecomposability uses its two nonisomorphic coordinate socles and the mixed-vector calculation; the opposite conclusion uses the two copies of the short branch top.
The final fiber-kernel contradiction rules out a nonuniserial left branch once the two spanning radical branches have simple intersection.
With simple nonzero intersection, both spanning radical branches are uniserial. The second conclusion is the first after interchanging the two branches.
The nonzero radical-square branch of the local induction is complete: the radical-top construction supplies two local branches spanning the radical. If they are disjoint, induction makes both uniserial directly; if they meet, the intersection obstruction makes the intersection simple and the final fiber-kernel obstruction makes both branches uniserial.
In the square-zero branch, the only remaining input needed for biseriality is the length bound on the radical. Square-zero makes the radical semisimple, and a semisimple radical of length at most two splits into at most two simple (hence uniserial) summands.
The radical-square-zero branch of the local induction. The radical is
semisimple, while the D₄ obstruction bounds its length by two.
One local step of the direct biserial induction.
Direct Pogorzały--Skowroński biserial induction: under a complete coordinate family, if every indecomposable finitely generated module is coordinate-thin, then every finitely generated module with simple top is biserial.
A primitive principal right projective has simple top.
Every primitive principal right projective is biserial when all indecomposable right modules are coordinate-thin.