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Let $A$ be a finite dimensional algebra of finite global dimension given by connected quiver and relations.

Do we have $gldim(A) \geq \min \{ \text{injdim}(S)+\text{projdim}(S) | S$ simple $\} $ in general? (edit: I just realised that the answer is probably no for acyclic quivers when we choose $injdim(S)=0$ and only short arrows start at the point at $S$. So the questions might be more interesting for algebras of finite global dimension with quivers without sinks or sources)

Does it hold for Nakayama algebras?

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