Trivial extension orders and dimension of singularity categories

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In this project, I try to understand under which conditions I can iterate the main result of A note on the global dimension of shifted orders. The main idea is that this can be done when "the Nakayama functor" on the category of centrally Cohen-Macaulay modules takes projective-injectives to projective-injectives. In the representation theory of finite dimensional algebras, such algebras are called Morita orders. Maybe, I will call these Morita orders or Morders?

Draft available here, comments welcome: