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Let R be a commutative ring, S an fí-algebra, H a Hopf Ralgebra, both finitely generated and projective as /¿-modules, and suppose S is an //-object, so that H* = Hotcir(H, R) acts on S via a measuring. Let / be the space of left integrals of //*. We say S has normal basis if S = H as H*modules, and S has local normal bases if Sp = Hv as //'-modules for all prime ideals p of R. When R is a perfect field, H is commutative and cocommutative, and certain obvious necessary conditions on S hold,doi:10.2307/2000648 fatcat:6rahounyubdubp6johmgxb7kpq