Let |$(\mathcal{K},\mathcal{O},k)$| be a |$p$|-modular system where |$p$| is a prime and |$k$| algebraically closed, let |$b$| be a |$G$|-invariant block of the normal subgroup |$H$| of a finite group |$G$|⁠, having defect pointed group |$Q_\delta$| in |$H$| and |$P_\gamma$| in |$G$| and consider the block extension |$b\mathcal{O}G$|⁠. One may attach to |$b$| an extended local category |$\mathcal{E}_{(b,H,G)}$|⁠, a group extension |$L$| of |$Z(Q)$| by |$N_G(Q_\delta )/C_H(Q)$| having |$P$| as a Sylow |$p$|-subgroup, and a cohomology class |$[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$|⁠. We prove that these objects are invariant under the |$G/H$|-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on |$p^{\prime}$|-extensions of inertial blocks.

中文翻译：

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区块扩展，本地类别和基本Morita等价

令| $（\ mathcal {K}，\ mathcal {O}，k）$ | 成为| $ p $ | 模块化系统，其中| $ p $ | 是素数和| $ k $ | 代数关闭，让| $ b $ | 成为| $ G $ | 子组的-不变块| $ H $ | 有限组| $ G $ |⁠的集合，具有缺陷指出组| $ Q_ \ delta $ | 在| $ H $ |中 和| $ P_ \ gamma $ | 在| $ G $ |中 并考虑块扩展| $ b \ mathcal {O} G $ |⁠。一个可以附加到| $ b $ | 扩展的本地类别| $ \ mathcal {E} _ {（b，H，G）} $ |⁠，组扩展| $ L $ |的| $ Z（Q）$ | 由| $ N_G（Q_ \ delta）/ C_H（Q）$ | 具有| $ P $ | 作为Sylow | $ p $ | -subgroup，以及H ^ 2（N_G（Q_ \ delta）/ QC_H（Q），k ^ \ times）$ |⁠中的同调类| $ [\ alpha] \。我们证明在| $ G / H $ |下这些对象是不变的。分级的基本森田等值。在此过程中，我们给出了Külshammer和Puig（1990）以及Puig和Zhou（2012）关于幂等块扩展的结果的替代证明。我们还通过方法推导了周（2016）关于| $ p ^ {\ prime} $ |的结果。-惯性块的延伸。