Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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For X (φi ) ∈ Morn(A-mod), def<strong>in</strong>e Ω−1X (φi ) to be<br />
�<br />
Ω−1 �<br />
X1<br />
.<br />
Ω −1 Xn<br />
(Ω −1 φ i )<br />
Proposition 3.2<br />
∈ Morn(A-mod).<br />
Let A be a self<strong>in</strong>jective algebra, X (φi ) ∈ Sn(A). Then there are the<br />
follow<strong>in</strong>g isomorphisms <strong>in</strong> Morn(A-mod)<br />
(i) τ j<br />
S X (φ i ) ∼ = τ j Rot j X (φi ) for j ≥ 1. In particular, τSX (φi ) ∼ = τ CokX (φi ).<br />
(ii) τ s(n+1)<br />
S X (φi ) ∼ = τ s(n+1) Ω−s(n−1) X (φi ), ∀ s ≥ 1.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 17 / 24