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We prove a finiteness theorem for the first flat cohomology group of finiteflat group schemes over integral normal proper varieties over finite fields. Asa consequence, we can prove the invariance of the finiteness of theTate-Shafarevich group of Abelian schemes over higher dimensional bases underisogenies and alterations over/of such bases for the $p$-part. Along the way,we generalize previous results on the Tate-Shafarevich group in this situation.

On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields