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For which Artin algebras $A$ is the stable module (that is the module category modulo projectives) category of finitely generated modules abelian? If you like you may take rings that are not Artin algebras or look at all modules instead of just finitely generated modules.

Non-trivial examples should include $A=K[x]/(x^2)$ or $A$ being a hereditary algebra of Dynkin type $A_2$ or $A_3$ with linear oriented orientation. I do not know many examples but all had dominant dimension at least one. Is there one with dominant dimension zero? Is there a nonselfinjective (and connected) example of infinite global dimension?

Added question (wild guess, june 2019): Is the stable module category abelian if and only if the stable Auslander algebra is isomorphic to an Auslander algebra (or has global dimension at most 2)?

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    $\begingroup$ If $A$ is not self-injective, then the stable module category is not triangulated, so the fact that triangulated abelian categories are semisimple only answers the question for self-injective algebras. $\endgroup$ Jan 4, 2018 at 10:19
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    $\begingroup$ By the way, don’t the $A_2$, $A_3$ examples extend to linearly ordered $A_n$ for every $n$? The natural inclusion of $\text{mod-}A_{n-1}$ (omitting the sink vertex) induces an equivalence from the module category of $A_{n-1}$ to the stable category of $A_n$. $\endgroup$ Jan 4, 2018 at 11:23
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    $\begingroup$ @მამუკაჯიბლაძე The stable categories (modulo projectives and modulo injectives) are used quite extensively in the representation theory of finite dimensional algebras. A good place to start might be the book “Representation theory of artin algebras” by Auslander, Reiten and Smalø. $\endgroup$ Jan 4, 2018 at 18:01
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    $\begingroup$ @მამუკაჯიბლაძე The class of Gorenstein algebras generalise selfinjective algebras and the stable category of Gorenstein projective modules is triangulated. A result by Rickard (I think) states that the stable category of a selfinjective algebra is equivalent to the singularity category and the generalisation (by Buchweitz I think) of this is that the stable category of Gorenstein projective modules is equivalent to the singularity category. $\endgroup$
    – Mare
    Jan 4, 2018 at 18:09
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    $\begingroup$ @მამუკაჯიბლაძე At least for Artin algebras or finite dimensional algebras there is a very large literature on that. Look up for Auslander-Reiten theory. A celebrated result is that the Auslander-Reiten translations establish an equivalence between the stable category (modulo projecitves) and the costable category modulo injectives. My favorite books on those subjects are the books by Skowronski and Yamagata called Frobenius algebras I and II (despite the title the books are about general finite dimensional algebras). Maybe an interesting question is what happens modulo projective+injectives? $\endgroup$
    – Mare
    Jan 4, 2018 at 18:12

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