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(Note: I modified the question and instead of looking at simple modules in the question, I look at all indecomposable modules.)

Let $A$ be a finite dimensional algebra over a field $K$ given by an acyclic quiver with relations and assume for simplicity that it has finite global dimension and the field is algebraically closed. It is well known that the projective dimension of the regular module as a bimodule equals the global dimension of the algebra. But what is the injective dimension of the regular module as a bimodule? It seems noone has studied that.

Question:

Do we have that the injective dimension of $A$ as an bimodule is equal to $\max \{ projdim(M)+injdim(M) | M$ is an indecomposable right A-module $ \}$ ?

If not, do we at least have that the injective dimension of $A$ is bounded by $\max \{ projdim(M)+injdim(M) | M$ is an indecomposable right A-module $ \}$ ?

It is true for all Nakayama algebras with a linear quiver and at most 6 simple modules, which are altogether 64 algebras (where the value of the injective dimension varies quite alot).

(By the way is the injective dimension of $A$ equal to the projective dimension of $D(A)$ as a bimodule? Im confused about this)

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