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Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$. Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal projective resolution of $M$, we get an exact sequence $0 \rightarrow M^{*} \rightarrow P_0^{*} \rightarrow ... \rightarrow P_{n-1}^{*}$ Here $(-)^{*}=Hom_A(-,A)$. In case $n=2$, the cokernel of the map $P_0^{*} \rightarrow P_1^{*}$ is the well known Auslander-Bridger transpose $Tr(M)$ of $M$.

Question: Is there a name or nice expression for the cokernel $P_{n-2}^{*} \rightarrow P_{n-1}^{*}$ for general $n \geq 3$?

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    $\begingroup$ Isn't your $\mathrm{Tr}(M)$ what people call the Auslander--Reiten transpose? Or am I missing something? (Perhaps just different nomenclature?) $\endgroup$
    – Pedro
    Commented May 6, 2020 at 11:37
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    $\begingroup$ @PedroTamaroff I think Tr is called Auslander-Bridger transpose (which exists for any noetherian ring) while $\tau$=DTr is called Auslander-Reiten translate for Artin algebras. $\endgroup$
    – Mare
    Commented May 6, 2020 at 11:39
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    $\begingroup$ Ah! I see. Thanks! :) $\endgroup$
    – Pedro
    Commented May 6, 2020 at 11:49
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    $\begingroup$ You probably already know that the cokernel of $P_{n-2}^{*} \rightarrow P_{n-1}^{*}$ is $\text{Tr }\Omega^{n-2}M$. I don't know a name, but Auslander and Bridger gave it a symbol, $J_{n-2}M$. $\endgroup$
    – Jeremy Rickard
    Commented May 6, 2020 at 14:20
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    $\begingroup$ If you apply the duality to $\operatorname{Tr}\Omega^{n-2}(M)$, then you get the $(n-1)$-Auslander-Reiten translation of $M$ (see sciencedirect.com/science/article/pii/S0001870806001721). $\endgroup$
    – Oeyvind Solberg
    Commented May 7, 2020 at 6:17

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