algebraic local charts

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

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I think the distinction between local charts and local local charts is superfluous ... – Martin Brandenburg Jan 27 '11 at 13:46
The examples that immediately comes to my mind are charts that are adapted to the situation at hand. For instance an affine cover over which a given locally free module is actually free. This happens of course also in differential or complex geometry but there is a difference in that for many purposes balls are enough and hence can be chosen once and for all in these latter cases. Hence I agree with Martin. – Torsten Ekedahl Jan 27 '11 at 13:56
in analytic case you have much more instruments like the inverse function theorem which you do not have in the algebraic case. You can overlap this using completions and then try to make the result algebraic, so I am asking if there is some rule to use this stuff. – unknown Jan 27 '11 at 14:11
In short, the answer is yes, if you use the etale topology. There is an inverse function theorem with respect to the etale topology. There is also a fairly general principle that you can lift solutions in the completion to solutions in an etale neighborhood (Artin approximation). A good toy example (and exercise!) is the fact that the map $C^* \to C^*$ given by $z \mapsto z^n$ becomes a local isomorphism in the etale topology. – Arend Bayer Jan 28 '11 at 11:13

1 Answer

What you're looking for is the étale topology (I think).

The fact is that Zariski opens are way too big to grasp stuff which should be more local than "Zariski local", for example any two smooth points on two $n$-dimensional varieties over $\mathbb{C}$ have isomorphic completed local rings (power series ring in $n$ indeterminates), but you will never find two isomorphic zariski open neighborhoods, unless the varieties are birational!

The way to go more locally is to consider étale morphisms to a scheme (which should morally be locally invertible, but they aren't in the algebraic context) to be "open subsets" of that scheme. Of course this doesn't make sense unless you modify the notion of topology, and in fact this is what led to the notion of Grothendieck topologies.

A good introduction to this stuff is Milne's book "étale cohomology", first chapters (I hope this is what you were looking for).

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