Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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The rotation of X (φi)<br />
The rotation RotX (φi ) of X (φi ) is def<strong>in</strong>ed to be<br />
(X1<br />
ψn−1<br />
��Y<br />
1<br />
n<br />
��<br />
· · · ψ1<br />
��Y<br />
1<br />
2 ) ∈ Morn(A-mod)<br />
(here,a for convenience we write the rotation <strong>in</strong> a row). We remark that<br />
RotX (φi ) is well-def<strong>in</strong>ed.<br />
Lemma 3.1<br />
Let X (φi ) ∈ Morn(A). Then RotX (φi ) ∼ = Cok MimoX (φi ) <strong>in</strong> Morn(A-mod).<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 16 / 24