Duality for Hopf Algebroids,

Our goal is twofold. First, we formulate a duality between commutative bialgebroids and cocommutative bialgebras over a ring extension. Second, we show that for a certain action groupoid J, the Hopf algebroid of functions on the Frobenius kernel J1 is dual to the restricted enveloping algebra of the Lie algebroid of J.