Left-right projective bimodules and stable equivalences of Morita type (Q2717733)

Given two finite-dimensional \(K\)-algebras \(A\) and \(B\), an \(A\)-\(B\)-bimodule \(X\) is called left-right projective if it is projective as a left \(A\)-module and as a right \(B\)-module. By \(\text{mod}(A)\) is denoted the category of all finite-dimensional right \(A\)-modules and by \(\underline{\text{mod}}(A)\) is denoted its stable category modulo projectives. For the enveloping algebra \(A^e=A^o\otimes_kA\), \(\text{lrp}(A^e)\) is the full subcategory of \(\text{mod}(A^e)\) consisting of the left-right projective \(A\)-bimodules and \(\underline{\text{lrp}}(A^e)\) is the stable category of \(\text{lrp}(A^e)\).NEWLINENEWLINENEWLINEThe following conditions are equivalent: (1) \(A\) and \(B\) are stably equivalent of Morita type; (2) There are left-right projective bimodules \(_AN_B\), \(_BM_A\) such that the functor \(M\otimes_A-\otimes_AN\colon\text{lrp}(A^e)\to\text{lrp}(B^e)\) induces an equivalence \(F\colon\underline{\text{lrp}}(A^e)\to\text{lrp}(B^e)\) with \(F(A)\cong B\) whose quasi-inverse is induced by the functor \(N\otimes_B-\otimes_BM\colon\text{lrp}(B^e)\to\text{lrp}(A^e)\).NEWLINENEWLINENEWLINEIn the case when \(A\) and \(B\) are self-injective finite-dimensional \(K\)-algebras, the connected component \(C\) (and its stable part \(C^s\)) of the Auslander-Reiten quiver \(\Gamma_{A^o\otimes_KB}\) of \(A^o\otimes_KB\) is considered and how many left-right projective vertices it contains is indicated. It is shown also for a self-injective finite-dimensional \(K\)-algebra \(A\) that the complement \(\text{mod}(A^e)\setminus\text{lrp}(A^e)\) is representation-infinite except for some trivial cases.