Auslander-Reiten Translations in Monomorphism Categories

Theorem 4.1
Let A be a selfinjective algebra, and FS be the Serre functor of Sn(A).
Then we have an isomorphism in Sn(A) for X (φi ) ∈ Sn(A) and for s ≥ 1
F s(n+1)
S X (φi ) ∼ = Mimo τ s(n+1) Ω −2sn X (φi ). (4.4)
Moreover, if d1 and d2 are positive integers such that τ d1M ∼ = M and
Ω d2M ∼ = M for each indecomposable nonprojective A-module M, then
F N(n+1)
S
X (φi ) ∼ d1
= X (φi ), where N = [ (n+1,d1) ,
d2
(2n,d2) ].
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 22 / 24