Given a property of the complete local ring of a variety at a point, how can we show that the set of all points on the variety sharing the same property is open or closed in the Zariski topology? Such questions are ubiquitous especially in resolution of singularities. In the example of the normal crossings property and some variations we give a solution via étale neighbourhoods and Artin approximation.

We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.