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Let $A$ be an algebra of finite global dimension and $Af$ the direct sum of all indecomposable projective-injective left $A$-modules. Using right modules, left $g$ denote the injective dimension of $A/AfA$.

Is it true that $g=\sup \{ injdim(S)-domdim(S) | S \ $simple $\}$?

I tested it for various algebras having dominant dimension at least one and found no counterexample.

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