Lifting smooth algebras and their morphisms
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.
- 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?