Abstract
Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of B. As an application, we obtain Maschke-type theorems for weak Hopf–Galois extensions and weak smash products.
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The research is supported by the National Natural Science Foundation of China (10871170) and the Fundamental Research Fund for the Central Universities (KYZ201125).
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Zhou, XY. Homological dimension of weak Hopf–Galois extensions. Acta Math Hung 138, 140–146 (2013). https://doi.org/10.1007/s10474-012-0229-0
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DOI: https://doi.org/10.1007/s10474-012-0229-0