Hi, in differential geometry or in complex geometry one of basic stuff to prove something is to do it on the local charts and then to check that the construction glues with the others charts. Which is the anologous of local charts (ball in $C^n$) in algebraic geometry (not only complex algebraic geometry) which are "more small" or more local then the affine schemes?

The question arise from the fact that many time I see that something is proved assuming that these local charts are Spec of some complete ring.

Thank you in advice