Expositiones Mathematicae ( IF 0.7 ) Pub Date : 2020-02-10 , DOI: 10.1016/j.exmath.2020.01.003 Lars Kadison
The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1 (centrally projective extension). Separable and split extensions have separability idempotents and bimodule projections in 1–1 correspondence via an endomorphism ring theorem in Section 3. If the separable idempotent is unique, then the separable extension is called uniquely separable. A Frobenius extension with invertible -index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier group-theoretic results are recovered and related to depth 1. The dual notion, uniquely split extension, only occurs trivially for finite group algebra extensions over complex numbers.
中文翻译:
唯一可分离的扩展
当在正确的内同态环中观察时,非交换环的可分离扩展的可分离张量元素是幂等的。因此,人们谈到可分离性幂等,就像可分离代数通常所说的那样。已经证明,仅当H深度为1(H可分离扩展)时,该幂等才是充满的。同样,拆分扩展具有双模块投影;当且仅当环延伸的深度为1(向心投影的延伸)时,该幂幂才满。通过第3节中的内同态环定理,可分离扩展和拆分扩展具有1-1对应关系的可分离等幂和双模投影。如果可分离幂等是唯一的,则可分离扩展称为唯一可分离。可逆的Frobenius扩展如果扶正器等于上环的中心,则-index是唯一可分离的。还表明,具有可逆E指数的半简单复数的唯一可分离扩展具有深度1。早期的组理论结果得以恢复并与深度1有关。对偶概念(唯一分裂扩展)仅对于有限组代数平凡地发生。复数的扩展。