Stable equivalence between universal covers of trivial extension self-injective algebras
Takayoshi Wakamatsu · Tsukuba Journal of Mathematics · 1985
In order to relate the categories mod-\^A and mod-R, Hughes-Waschb\"usch used the exact functor $\Phi:m\propto 1-\hat{A}\rightarrow mod-R$ which preserves the indecomposability and the composition length of a module and also almost split sequences and ir- reducible maps.Similarly to the functor $\Phi$ , we can define the functors $\Phi_{n}$ : $mod-\hat{A}\rightarrow mod-R_{n}$ and $\Phi_{m.n}$: $mod-R_{m\cdot n}\rightarrow mod-R_{n}$ .We shall show that the functors $S_{1}=S,$ $S_{2},$ $S_{3},$ $\cdots,$ $S_{\infty}$ make the following diagrams commutative:It should be noted that the functor $\Phi$ is not dense in general, though in the case where $R$ is representation-finite or $A$ is hereditary $\Phi=\Phi_{1}$ is dense andRecently, D. Happel [15] has proved that mod-\^A and mod-B are equivalent if gl. $dim$ .$ A<\infty$ .But, since $\Phi$ is not dense in general even if gl. $dim$ .$ A<\infty$ , our results does not follow from his one.At the end of this paper such an example will be given.Throughout this paper, we fix a commutative artin ring $K$ and all algebras are assumed to be artin K-algebras except $R_{\infty}$ and $S_{\infty}$ , and modules are finitely generated over $K$ and morphisms operate on the opposite side of the scalars.The ordinary duality functor is always denoted by $D$ .