Genealogical Geometry – Theorem on formal functions and cohomological flatness

Leave $ f: X rightarrow S $ Be an adequate morphism of schemes with a Noetherian objective. The theorem on formal functions says that for any point $ s in S $ There is an isomorphism between the inverse limits of $ (f_ * O_X) _s / mathfrak {m} _s ^ n (f_ * O_X) _s $ Y $ Gamma (X_s, O_X otimes_ {O_S} O_S / mathfrak {m} _s ^ n O_S) $, if I understand correctly. Yes
$ f $ happens to be flat and cohomologically flat in degree 0, then we know that the isomorphism between the inverse limits is actually an isomorphism in $ n $-th stage for any $ n> 0 $.

  • Is there an example of a morphism? $ f $ in such a way that the induced comparison morphism is an isomorphism in $ n $-a stage (and therefore isomorphism in $ m $-th stage for each $ 0 <m <n $) And is not it an isomorphism at a later stage?
  • Is there a description of $ (f_ * O_X) _s $ in terms of fiber information?