Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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Applications to self<strong>in</strong>jective algebras<br />
A: a self<strong>in</strong>jective Art<strong>in</strong> algebra,<br />
A-mod: the stable category of A-mod<br />
Morn(A-mod): the morphism category of A-mod<br />
�<br />
Objects: X (φi ) =<br />
⎛<br />
f1<br />
� X1<br />
⎞<br />
.<br />
Xn<br />
(φi )<br />
, φi : Xi+1 → Xi <strong>in</strong> A-mod,<br />
Morphisms: ⎝ ⎠ : X (φi .<br />
) → Y (θi ), fi : Xi → Yi such that the follow<strong>in</strong>g<br />
fn<br />
diagram commutes <strong>in</strong> A-mod<br />
fn<br />
Xn<br />
��<br />
Yn<br />
φn−1<br />
θn−1<br />
��<br />
Xn−1<br />
fn−1<br />
��<br />
��<br />
Yn−1<br />
φn−2<br />
θn−2<br />
��<br />
· · ·<br />
��<br />
· · ·<br />
φ2<br />
θ2<br />
��<br />
X2<br />
f2<br />
��<br />
��<br />
Y2<br />
φ1<br />
θ1<br />
��<br />
f1<br />
X1<br />
��<br />
��<br />
Y1.<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 14 / 24