Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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Remark 2.2<br />
The process means:<br />
Give an object X (φi ) <strong>in</strong> Sn(A)<br />
τSX (φi ) ∼ = Mimo τ CokX (φi ). (1)<br />
Take the cokernel object X ′ (φ ′ i ) = CokX (φ i ).<br />
Apply τ to these maps φ ′ i (1 ≤ i ≤ n − 1).<br />
Represent τCokX (φi ) by an object X ′′<br />
(φ ′′<br />
i<br />
) =<br />
� X ′′<br />
1<br />
.<br />
X ′′<br />
n<br />
�<br />
(φ ′′<br />
i )<br />
where X ′′<br />
1 , X ′′<br />
2 , · · · , X ′′<br />
n−1 have no nonzero <strong>in</strong>jective direct<br />
summands.<br />
Apply Mimo, there is a well-def<strong>in</strong>ed object <strong>in</strong> Sn(A) up to<br />
isomorphism.<br />
<strong>in</strong> Morn(A)<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 11 / 24
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