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Let $H$ be a hereditary (connected?) finite-dimensional $k$-algebra. Then for any finite-dimensional modules $X, Y$ over $H$ one has functorial isomorphisms

\begin{equation*} \text{Ext}^{1}_H(X, Y) \cong D \text{Hom}_{H}(Y, \tau (X)) \cong D \text{Hom}_H (\tau^{-}(Y), X), \end{equation*}

where $\tau$ and $\tau^{-}$ are the Auslander-Reiten translations.

Question: Do you know a citable reference for this version of the Auslander-Reiten formula?

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If you are willing to cite a textbook, then the one of Assem, Simson and Skowroński, Elements of the representation theory of associative algebras. Vol. 1, has the following result in Chapter IV:

2.14. Corollary. Let $A$ be a $K$-algebra and $M$, $N$ be two modules in $\operatorname{mod}A$.

(a) If $\operatorname{pd} M \leq 1$ and $N$ is arbitrary, then there exists a $K$-linear isomorphism $$ \operatorname{Ext}^1_A(M,N) \cong D\operatorname{Hom}_A(N, \tau M). $$

(b) If $\operatorname{id} N \leq 1$ and $M$ is arbitrary, then there exists a $K$-linear isomorphism $$ \operatorname{Ext}^1_A(M,N) \cong D\operatorname{Hom}_A(\tau^{-1}N, M). $$

Since the conditions $\operatorname{pd} M \leq 1$ and $\operatorname{id} N \leq 1$ are always satisfied over a hereditary algebra, this implies the result you are looking for.

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