Lifting smooth algebras and their morphisms

Alberto Arabia

Let R be any commutative ring with unit an let I be any ideal of R. By reduction modulo I one gets a functor from the category of R-algebras to the category of R/I-algebras. This functor transforms a smooth R-algebra into a smooth R/I-algebra. The paper gives positive answers to the following inverse questions:

As a corollary very smooth liftings of a given smooth F_p-algebra always exist; which is a fondamental question in Monsky-Washnitzer cohomology theory.