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Recently I am reading the paper "On the stable module category of a self-injective algebra", the link is here: http://www.ams.org/journals/tran/2000-352-05/S0002-9947-00-02232-7/S0002-9947-00-02232-7.pdf

There are two places I don't know:

  1. At page 2391, 1.2 says $\Omega$ induces an equivalent of the stable category of $\Lambda$, and so does the Auslander-Reiten translate $\tau$. In particular, $\Omega$ induces a graph isomorphism of the stable Auslander-Reiten quiver of $\Lambda$.
  2. At page 2391, 1.3 lists three functorial isomorphisms.

I am not familar with the first place and don't know the proof of the second place. Is there anyone who can describe these more clear or provide me some relevant materials? Thank you very much.

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The book "Frobenius algebras I" by Skowronski and Yamagata has a reference for 1. in chapter IV. 8. and the functorial isomorphism are the Auslander-Reiten formulas which can be found in chapter III. theorem 6.3. in the same book. The thing with the graph isomorphism is also explained in the book by Auslander, Reiten and Smalo.

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