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edited Oct 4, 2018 at 9:41

user129156

ag.algebraic-geometry ac.commutative-algebra arithmetic-geometry deformation-theory etale-cohomology formal-schemes

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edited Oct 4, 2018 at 9:35

user129156

ag.algebraic-geometry ac.commutative-algebra arithmetic-geometry etale-cohomology formal-schemes

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asked Oct 4, 2018 at 9:26

user129156

Smooth loci and formal neighborhoods

Let $R$ be a Noetherian local ring with maximal ideal $I$.

Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$

$$f_n : A/I^n \to B/I^n$$

is an isomorphism for all $n\ge 1$. What can we say about $f$? Is $f$ smooth?

ag.algebraic-geometry ac.commutative-algebra arithmetic-geometry etale-cohomology