Auslander-Reiten Translations in Monomorphism Categories

Theorem 3.3
Let A be a selfinjective algebra, and X (φi ) ∈ Sn(A). Then we have
τ s(n+1)
S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ), s ≥ 1. (2)
Applying the above theorem to the selfinjective Nakayama algebras
A(m, t), we get
Corollary 3.4
For an indecomposable nonprojective object X (φi ) ∈ Sn(A(m, t)),
m ≥ 1, t ≥ 2, there are the following isomorphisms:
(i) If n is odd, then τ m(n+1)
S X (φi ) ∼ = X (φi );
(ii) If n is even, then τ 2m(n+1)
S X (φi ) ∼ = X (φi ).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 18 / 24