Projective-radical recursion under the beta-two bound #
This file implements the finite radical recursion in Auslander--Reiten, Lemma 4.5. The common left/right beta bound makes the radical of a projective occurring irreducibly inside a noninjective projective either zero or indecomposable. If that radical is nonprojective, Proposition 1.3 makes it uniserial; if it is projective, the argument repeats at the strictly smaller radical.
Over an algebraically closed field, the dimension of the intrinsic irreducible-morphism space is the official arrow multiplicity, at projective and nonprojective endpoints alike.
An actual irreducible morphism forces positive official arrow multiplicity.
Positive incoming multiplicity at a nonprojective endpoint supplies a literal occurrence in its selected minimal right almost-split middle.
If a projective embeds irreducibly in a noninjective projective, its projective-boundary radical has at most one indecomposable occurrence.
Under a global two-middle-term bound, a projective which occurs irreducibly inside another projective has radical arity at most one. The projective occurrence itself uses one of the two places in the translated almost-split middle.
Finite projective-radical recursion. A projective source of an irreducible morphism to a projective is uniserial under the global two-middle-term bound.
A global two-middle-term bound gives at most two indecomposable summands in the radical of a noninjective indecomposable projective.
Every displayed indecomposable summand of a noninjective projective's radical is uniserial under the global two-middle-term bound.
A global right-middle arity bound is self-dual. Contragredient duality reverses arrows, while inverse Auslander--Reiten translation rewrites the resulting outgoing sum as an incoming right-middle sum in the original skeleton.
If the nonprojective right meshes have arity at most two and a chosen projective has projective-boundary arity at most two, then its radical is the internal direct sum of at most two uniserial branches.
Under the global two-middle-term bound, the radical of every noninjective indecomposable projective is the internal direct sum of at most two uniserial branches.
A total right-middle arity bound of two makes every indecomposable projective radical an internal direct sum of at most two uniserial branches.
Under the global two-middle-term bound, every noninjective indecomposable projective is biserial.
Under a total two-middle-term bound, every indecomposable projective is biserial.