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Mar 8 at 17:29 comment added Jason Starr I see. The OP listed different statements in the post as (2). Thank you for clarifying.
Mar 8 at 1:01 comment added afh @Jason Starr I am referring to (2) in the questions (at the end of the post). The claim is that $R^0f_*(\mathcal{O}_X)$ is locally free and its formation commutes with base-change.
Mar 7 at 17:20 comment added Jason Starr I am not certain what you are claiming about (2). Clearly (2) fails in mixed characteristic and positive characteristic because of supersingular Enriques surfaces (among many other such smooth projective schemes).
Mar 7 at 0:07 vote accept Ben C
Mar 6 at 22:57 history answered afh CC BY-SA 4.0