A note on the global dimension of shifted orders

I consider dominant dimension of an order over a Cohen-Macaulay local ring inside the category of centrally Cohen-Macaulay modules. I follow methods of Matthew Pressland and Julia Sauter to prove that if the global dimension is large enough and the dominant dimension is positive, then a canonical tilted algebra (called the shifted order here) has strictly smaller global dimension when the Krull dimension of the base ring is positive.

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