Artin approximation and etale local structure
If two pointed varieties (X, x) and (Y, y) have isomorphic complete
local rings, then they have isomorphic elementary etale neighbourhoods.
More interesting result: Given (X, x) ---> (S, s) and (Y, y) ---> (T, t)
then if
O_{S, s}^ ---> O_{X, x}^
is isomorphic to a completed base change of
O_{T, t}^ ---> O_{Y,y}^
by a map O_{T, t}^ ---> O_{S, s}^ then there is an elementary etale
neighbourhood (V, v) of (S, s) and a morphism (V, v) ---> (T, t) such
that the pullback of Y by this morphism is etale locally isomorphic to
X over S... In fact what we prove is even a little bit stronger...
Also it is quite annoying and not very readable I'm afraid.
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