Let $A$ be a finite dimensional $k$-algebra, $k$ is a field, with a finite global dimension. I wonder if that implies $A$ is tame or finite type? or more generally is there a relation between these two notions?
2025-01-13 12:07:19.1736770039
Finite dimensional algebras with finite global dimension.
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This is really expanding on egreg's comment: there are wild algebras of any finite global dimension. Indeed, if you take $A$ to be a wild hereditary algebra (these exist: take the path algebra of any connected non-Dynkin or non-extended-Dynkin quiver), and $B$ to be any algebra of global dimension $n$, then the algebra $A\times B$ is wild of global dimension $n$.