# 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 $$0 ) And is not it an isomorphism at a later stage?
• Is there a description of $$(f_ * O_X) _s$$ in terms of fiber information?