Branch placement in a coordinate-thin biserial sum #
After quotienting the intersection of two submodules, their sum is the product of the two branch images. Coordinate thinness then prevents a simple-top submodule of the sum from projecting nontrivially to both factors.
A branch regarded as a submodule of the sum of two branches.
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The other branch regarded as a submodule of the sum.
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The branch intersection regarded inside the branch sum.
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In the quotient of a branch sum by the branch intersection, the two branch images are complementary.
The quotient of a two-branch sum by the branch intersection is the product of the two branch images.
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The range of a map from a simple-top module into a coordinate-thin sum of two branches lies in one branch. Modulo the branch intersection the sum is a product; nonzero projections to both factors would repeat a coordinate of the simple top.
The range of a map from a simple-top module into the radical of a coordinate-thin biserial module is uniserial. The map need not be injective: branch placement is applied to its range in the target.
The image of a simple-top radical submodule in a coordinate-thin biserial quotient is uniserial.
Images of two submodules under a map are comparable when the full range of the map is uniserial.
Two disjoint submodules remain disjoint after quotienting the ambient module by its socle. An element that could identify the two images would give a socle element in their direct sum, whose two coordinates already lie in the intrinsic socles and hence vanish in the ambient socle quotient.
In a coordinate-thin module, two submodules whose intersection lies in the socle have disjoint images after quotienting by that socle. Otherwise a nonzero coordinate in the common quotient image would occur in both submodules, while their actual intersection has zero contribution in that coordinate.
Submodules contained in one uniserial submodule are comparable in the ambient module.
If two uniserial branches lie in a submodule whose image in the ambient socle quotient is uniserial, then a terminal branch has only a simple, disjoint companion branch. The companion vanishes in the socle quotient and can therefore be removed without changing the terminal branch.
A nonsimple uniserial submodule properly contained in a local biserial side has one of the two carrier forms needed by the common-radical obstruction. Either it has an ambient uniserial immediate successor, or a simple companion branch can be quotiented out; in the latter case the resulting uniserial carrier has the same top as the original side.
A nonsemisimple intersection properly contained in a biserial side has a maximal nonsimple uniserial submodule in the ambient module. The biseriality hypothesis is needed only on the side, not on the ambient module whose coordinate thinness performs the branch placement.
Two coordinate-thin uniserial extensions of the same nonsimple uniserial radical give the common-radical diagonal-cokernel contradiction. This endpoint is independent of whether either extension was constructed as an ambient submodule or as a quotient carrier.
An immediate uniserial successor of a maximal nonsimple uniserial submodule meets any other containing submodule in exactly the chosen submodule, provided the intersection stays in the maximality region.
The complete carrier assembly for a nonsemisimple intersection of two biserial local sides. Each side supplies either an ambient immediate successor or a quotient carrier. Ambient intersections are controlled by maximality, while quotient-carrier tops are transported from the original two nonisomorphic side tops.