Irreducible maps with a repeated binary-biproduct summand #
When the two summands in the target of a binary lift are isomorphic, the off-diagonal-radical criterion does not apply. This file replaces it by the standard multiplicity-space criterion. Endomorphisms of the repeated indecomposable are assumed scalar modulo the categorical radical, and every scalar row operation on the two component maps is assumed irreducible.
The proof is Gaussian elimination modulo the radical. After removing the scalar part of one cross term, the corresponding linear combination is still irreducible and hence supplies a second splitting. The remaining cross terms are radical, so the resulting two-by-two matrix is the identity plus a radical morphism and is invertible.
Over an algebraically closed field, an endomorphism of a nonzero object with finite-dimensional local endomorphism ring is scalar modulo the categorical radical.
A repeated-summand binary lift is irreducible when scalar row operations on its two components remain irreducible and endomorphisms of the repeated summand are scalar modulo the radical.
The symmetric form of the repeated-summand lift criterion, with scalar row operations based at the second component.
A repeated-summand binary desc is irreducible when scalar column operations on its two components remain irreducible and endomorphisms of the repeated summand are scalar modulo the radical.
The symmetric form of the repeated-summand desc criterion, with scalar column operations based at the second component.
The repeated-summand lift criterion after identifying two isomorphic target summands.
The repeated-summand desc criterion after identifying two isomorphic source summands.