The initial complementary branch in an infinite Iyama right ladder #
This module integrates the dependent infinite special right ladder with the radical-power annihilator propagation from Iyama, Lemma 6.4.1(1)(i). It treats the zero-th padded summand separately. The argument is entirely categorical and contains no concrete algebra or module classification.
The weak-kernel mesh models stored in an infinite special right ladder give exactly the annihilator-propagation hypothesis used by the radical-power argument.
The connecting maps in an infinite special right ladder are radical.
Ladder-specific form of radical-power annihilator propagation.
The source-facing radical-power witness required in Iyama 6.4.1(1)(i).
Instances For
If the initial padded summand is nonzero, its split inclusion into the
source of a₀ is already a nonzero degree-zero witness.
If the initial padded summand is zero, a nonzero annihilator of a₀
transports across the initial arrow isomorphism to a nonzero annihilator of
the essential map b 0.
The complete U₀-aware radical-power conclusion.
Given a nonzero map annihilating the initial special arrow, some padded
summand U n admits a nonzero map back to the original source in the nth
power of the chosen categorical radical. The n = 0 case is supplied by
the initial padded complement; if that complement is zero, the existing
right-ladder propagation supplies a witness with index n + 1.
Direct existential form of the preceding theorem.
Paper-facing form: build the infinite ladder from a special initial arrow and then obtain the radical-power witness at its original source.
A nonmonic special arrow has a nonzero radical-power witness on one of the padded terms in its infinite right ladder. This is the finite nilpotent-radical specialization of the first conclusion in Iyama 6.4.1(1)(i).