Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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The object MimoX (φi)<br />
Def<strong>in</strong>e MimoX (φi ) to be the object<br />
⎛<br />
⎜<br />
⎝<br />
That is<br />
Xn<br />
X1⊕IKer φ1⊕···⊕IKer φn−1<br />
X2⊕IKer φ2⊕···⊕IKer φn−1<br />
.<br />
Xn−1⊕IKer φn−1<br />
Xn<br />
⎞<br />
⎟<br />
⎠<br />
θn−1 ��<br />
Xn−1 ⊕ IKer φn−1<br />
(θ i )<br />
⎛<br />
⎜<br />
where θi = ⎜<br />
⎝<br />
θn−2 ��<br />
· · ·<br />
φi 0 0 ··· 0<br />
ei 0 0 ··· 0<br />
0 1 0 ··· 0<br />
0 0 1 ··· 0<br />
. . . ··· .<br />
0 0 0 ··· 1<br />
⎞<br />
⎟<br />
⎠<br />
(n−i+1)×(n−i)<br />
θ1 ��<br />
X1 ⊕ IKer φ1 ⊕ · · · ⊕ IKer φn−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 9 / 24<br />
.