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In this paper we set up the foundations around the notions of formaldifferentiation and formal integration in the context of commutative Hopfalgebroids and Lie-Rinehart algebras. Specifically, we construct acontravariant functor from the category of commutative Hopf algebroids with afixed base algebra to that of Lie-Rinehart algebras over the same algebra, thedifferentiation functor, which can be seen as an algebraic counterpart to thedifferentiation process from Lie groupoids to Lie algebroids. The other wayaround, we provide two interrelated contravariant functors form the category ofLie-Rinehart algebras to that of commutative Hopf algebroids, the integrationfunctors. One of them yields a contravariant adjunction together with thedifferentiation functor. Under mild conditions, essentially on the basealgebra, the other integration functor only induces an adjunction at the levelof Galois Hopf algebroids. By employing the differentiation functor, we alsoanalyse the geometric separability of a given morphism of Hopf algebroids.Several examples and applications are presented along the exposition.

Towards differentiation and integration between Hopf algebroids and Lie algebroids