Matthew Emerton,
The Artin Approximation Theorem
The Artin approximation theorem states that (in a suitable context)
one can always approximate formal power series solutions of algebraic
equations by convergent power series approximations (in the analytic
situation) or by algebraic solutions (in the algebro-geometric
situation).
Since formal computations are often substantially easier than algebraic
or analytic ones, this is a powerful tool in geometry for obtaining
local information about varieties.
We will present a proof of the theorem in the context of algebraic
varieties over an algebraically closed field, and discuss some applications.