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 morphisms a and b are homotopic?