# Why is Auslander correspondence a bijection between the set of Morita-equivalence classes?

The Auslander correspondence said there exists a bijection between the set of Morita-equivalence classes of representation-finite finite-dimensional algebras $$\Lambda$$ and that of finite-dimensional algebras $$\Gamma$$ with gl.dim $$\Gamma \leq 2 \leq$$ dom.dim $$\Gamma$$. It is given by $$\Lambda \mapsto \Gamma:=\operatorname{End}_{\Lambda}(M)$$ for an additive generator $$M$$ of $$\bmod \Lambda$$.

I don't understand why $$\Lambda \mapsto \Gamma:=\operatorname{End}_{\Lambda}(M)$$ is a bijection between the set of Morita-equivalence classes. And what is the importance of Morita-equivalence classes in representation theory of Artin algebra?

Thank you.

Let me answer your second question about the importance of Morita-equivalence classes first. Two rings $$R$$ and $$S$$ are called Morita equivalent if $$\operatorname{Mod} R$$ and $$\operatorname{Mod} S$$ are equivalent as categories. If one considers representation theory as the study modules over a ring, then in some sense $$R$$ and $$S$$ are indistinguishable, so it makes sense to study rings up to Morita equivalence. (Of course, there are many other aspects of representation theory that people are insterested in, which differ a lot if one changes the Morita representative: monoidal structure, dimension of simple modules, existence of subrings, just to name a few).
Now with your particular question, you said that the map is defined by sending $$\Lambda$$ to $$\operatorname{End}_\Lambda(M)$$ where $$M$$ is a additive generator of $$\operatorname{mod} \Lambda$$. If you would just define that on isomorphism classes, instead of Morita equivalence classes, then this would not be well-defined, as there are many different such additive generators, yielding non-isomorphic algebras. Just to give a trivial example: If $$R=\Bbbk$$ is the ground field, then $$\Bbbk^2$$ is an additive generator of $$\operatorname{mod}\Lambda$$ and $$\operatorname{End}_\Bbbk (\Bbbk^2)\cong \operatorname{Mat}_{2\times 2}(\Bbbk)$$ is Morita equivalent, but not isomorphic to $$\Bbbk\cong \operatorname{End}_\Bbbk(\Bbbk)$$. A different way to fix this uses the basic representative on each side. Then you would send (a basic representation-finite) $$\Lambda$$ to $$\operatorname{End}_\Lambda(M)$$ where $$M$$ has precisely one summand from each isomorphism class of indecomposable modules.