Auslander-Reiten Translations in Monomorphism Categories

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The notions A: an Artin algebra, A-mod: the category of all fin. gen. left A-modules Morn(A): the morphism category � of A-mod, n ≥ 2 Objects: X (φi ) = Xn � X1 . Xn (φ i ) φn−1 �� Xn−1 , φ i : X i+1 → X i are A-maps, i.e. φn−2 �� · · · � f1 φ2 �� X2 φ1 �� X1 Morphisms: f : X (φi ) → Y (θi ) is f = . , where fi : Xi → Yi are fn A-maps for 1 ≤ i ≤ n, such that the following diagram commutes fn Xn �� Yn φn−1 θn−1 � �� Xn−1 fn−1 �� Yn−1 φn−2 �� · · · θn−2 �� · · · � φ2 θ2 � �� X2 f2 �� Y2 φ1 θ1 � �� f1 X1 �� Y1. Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 3 / 24
The notions The monomorphism category Sn(A) is the full subcategory of Morn(A) consisting of all the objects X (φi ) where φ i : X i+1 −→ X i are monomorphisms for 1 ≤ i ≤ n − 1. Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 4 / 24
Page 1 and 2: Auslander-Reiten Translations in Mo
Page 3 and 4: Motivation C. M. Ringel and M. Schm
Page 5: The notions A: an Artin algebra, A-
Page 9 and 10: The kernel and cokernel functors Ke
Page 11 and 12: The functor: Mono Mono : Morn(A)
Page 13 and 14: The object MimoX (φi) Define MimoX
Page 15 and 16: Remark 2.2 The process means: Give
Page 17 and 18: An example The Auslander-Reiten qui
Page 19 and 20: Applications to selfinjective algeb
Page 21 and 22: The rotation of X (φi) The rotatio
Page 23 and 24: For X (φi ) ∈ Morn(A-mod), defin
Page 25 and 26: Theorem 3.3 Let A be a selfinjectiv
Page 27 and 28: An example Let X (φi ) be an indec
Page 29 and 30: Theorem 4.1 Let A be a selfinjectiv
Page 31: Thank you! E-mail: xiongbaolin@gmai

Auslander-Reiten Translations in Monomorphism Categories

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