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Let $A$ be a finite dimensional algebra over a field $k$ and global dimension of $A$ is finite. I want to study $A$ as a bimodule i.e. as $A^e=A \otimes A^{op}$-module. It is easy to see that $pd_{A^e} A=gl(A)=n$. My question is can one construct resolution of $A$ of length $n$ in some natural way? I mean, for example, bar resolution is given by some natural construction, but it is too big, Koszul resolution is another example of natural resolution (for commutative rings), perhaps there are other resolutions that, for example, uses information about structure of indecomposable projectives or some other facts about $A$.

Second question is there a construction of injective resolution of $A$ over $A$?

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Lemma 1.5 in [Happel: Hochschild cohomology for finite-dimensional algebras] gives the projectives modules appearing in the minimal projective resolution of $A$ as an $A^e$-module (but not the maps). – Julian Kuelshammer May 11 '13 at 8:14

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