Coordinate-thin modules #
For a family of idempotents in a finite-dimensional algebra, a finitely
generated right module is coordinate-thin when every space Xe has
coefficient-field dimension at most one. For a basic algebra and a complete
primitive family, these dimensions are the usual simple composition-factor
multiplicities. The definition below keeps only the coordinate statement
actually supplied by finite-category modules and used in the biseriality
argument.
Every chosen idempotent coordinate of a finitely generated right module has dimension at most one.
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Every indecomposable finitely generated right module is coordinate-thin. This is the algebraic form of the multiplicity-free premise used in the biseriality induction.
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A module isomorphism identifies the coordinates belonging to an idempotent.
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Coordinate thinness is invariant under module isomorphism.
A module map restricts to every idempotent coordinate.
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An injective module map remains injective on every idempotent coordinate.
A surjective module map remains surjective on an idempotent coordinate. The idempotent projects any chosen preimage back into that coordinate.
A positive idempotent coordinate remains positive under an injective module map.
A positive idempotent coordinate in a quotient was already positive in the source.
A submodule of a finitely generated right module, retained as an object of the finitely generated module category.
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Inclusion of a submodule restricts to an inclusion on every idempotent coordinate.
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The coordinate inclusion belonging to a module submodule is injective.
Coordinate thinness passes to finitely generated submodules.
A quotient of a finitely generated right module, retained as an object of the finitely generated module category.
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The quotient map, bundled with the finitely generated quotient wrapper as its codomain.
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The bundled quotient map is surjective.
A linear map killing a submodule descends to the finitely generated quotient wrapper.
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A surjective map remains surjective after it is descended across a submodule contained in its kernel.
The canonical map between two nested finitely generated quotients.
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Every canonical map between nested quotients is surjective.
The kernel of the canonical map M/P → M/Q is the image of Q.
The quotient map of a module restricts to a linear map on every idempotent coordinate.
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For an idempotent, the coordinate map induced by a module quotient is surjective.
Taking an idempotent coordinate preserves the canonical short exact sequence of a submodule and its quotient.
Idempotent-coordinate dimensions add across a submodule and its quotient. This is the composition-multiplicity additivity needed when two isomorphic simple factors occur in different Loewy layers rather than as a semisimple product subquotient.
Coordinate thinness passes to finitely generated quotients.
The product of two finitely generated right modules, retained as an object of the finitely generated module category.
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An idempotent coordinate of a binary product is the product of the two idempotent coordinates.
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A module has a repeated self-subquotient when some subquotient is the product of two copies of a nonzero finitely generated module.
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A surjection from W onto two copies of a nonzero module is a repeated
self-subquotient certificate.
A submodule which is linearly equivalent to two copies of one nonzero module is already a repeated self-subquotient certificate.
Two disjoint isomorphic nonzero submodules form a repeated submodule of the ambient module.
A submodule which surjects onto two copies of one nonzero module gives a repeated self-subquotient of the ambient module.
A submodule consisting of two copies of a nonzero module inside a
quotient of W gives a repeated self-subquotient of W. The witnessing
submodule is pulled back along the quotient map and then divided by the
kernel of the restricted surjection.
Coordinate dimensions add under binary products.
Two copies of a module with a nonzero chosen coordinate cannot form a coordinate-thin product.
A module is not coordinate-thin if one of its subquotients is two copies of a module having a nonzero coordinate. This is the reusable contradiction form for the repeated-simple-factor obstruction modules in the biseriality induction.
A nonzero module has a nonzero coordinate for some member of a complete orthogonal idempotent family.
In a coordinate-thin module, no nonzero submodule of the radical is isomorphic to the top. The submodule contributes a positive coordinate to the radical, the isomorphic top contributes the same positive coordinate to the quotient, and coordinate additivity would make the ambient coordinate at least two-dimensional.
A nonzero map out of a simple-top module makes every nonzero coordinate of the source top occur in the target. The kernel is proper, hence lies in the source radical, so the source top is a quotient of the image.
If a binary product is coordinate-thin, a simple-top module cannot map nontrivially to both factors. The same nonzero coordinate of the source top would occur in both factors and hence twice in their product.
In a coordinate-thin simple-top module, every map to a subquotient of a module embedded in its Jacobson radical is zero. A nonzero map would make a coordinate of the source top occur again inside the radical.
Submodule form of
linearMap_to_injective_jacobson_subquotient_eq_zero_of_coordinateThin.
Over a complete idempotent family, a subquotient consisting of two copies of any nonzero module is enough to refute coordinate thinness.
Under the all-indecomposables-thin premise, no indecomposable module can have a subquotient consisting of two copies of a nonzero module.
Complete coordinate thinness forbids repeated self-subquotients in every indecomposable module.