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We prove several results concerning the existence of potentially crystallinelifts with prescribed Hodge-Tate weights and inertial types of a givenn-dimensional mod p representation of the absolute Galois group of K, whereK/Q_p is a finite extension. Some of these results are proved by purely localmethods, and are expected to be useful in the application of automorphy liftingtheorems. The proofs of the other results are global, making use of automorphylifting theorems.

Potentially crystalline lifts of certain prescribed types