Two-term tilting complexes for selfinjective algebras

(1)引入自射代数的反射符号，确定与反射相关的Brauer树的变换。特别是，我们提供了一种将每个布劳尔树转换为布劳尔线的方法。(2)我们证明了在一个Artin代数上，每一个两项可倾复都有一个可倾模作为同调群在一个因子代数上。同时，我们确定了这种同态群的自同态代数，并将其作为二项倾斜复合体的自同态代数的一个因子代数给出。因此，由两项倾斜复形给出的Artin代数之间的每一个推导等价都可以推导出相应因子代数之间的推导等价。

(1) We introduce the notation of reflections for selfinjective algebras and determine the transformations of Brauer trees associated with reflections. In particular, we provide a way to transform every Brauer tree into a Brauer line. (2) We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras.