## 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:

- Is every smooth
*R/I*-algebra the reduction of a smooth *R*-algebra?
- In what extent given two
*R*-algebra morphisms *a,b:***A**-->**B** where *A* is smooth and such that the reductions *(a mod ***I**) and *(b mod ***I**) are homotopic *R/I*-algebra morphisms, the morphismes *a* and *b* are homotopic?

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.