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


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*Is there an example of a morphism $f$ such that the induced comparison morphism is an isomorphism at $n$-th stage (and thus isomorphism at $m$-th stage for every $0<m<n$) and is not an isomorphism at any later stage? 

*Is there a description of $(f_*O_X)_s$ in terms of fiberwise information?  

 A: Welcome new contributor.  With the hypothesis on flatness and cohomological flatness in degree $0$, there are isomorphisms at every stage.  Without this hypothesis, this can fail.  
Let $S$ be Spec of a local Noetherian ring $R$ with maximal ideal $\mathfrak{m}$.  Let $f:X\to S$ be a proper morphism.  Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.  For every integer $\ell\geq 0$, denote by $r_\ell=r_{\mathcal{F},f,\ell}$ the following $R$-module homomorphism, $$r_\ell:f_*\mathcal{F}/\left(\mathfrak{m}^\ell\cdot f_*\mathcal{F}\right) \to f_*\left( \mathcal{F}/\left(\mathfrak{m}^\ell\cdot \mathcal{F}\right) \right).$$
Definition 1. The sheaf $\mathcal{F}$ is $f$-typical to order $n$ if $r_\ell$ is surjective for $0\leq \ell < n$. 
The goal is to find a proper morphism $f$ such that the structure sheaf is $f$-typical to order $n$, but not to order $n+1$.
Lemma 2. If there exists a proper morphism $\pi:Y\to S$ and a coherent $\mathcal{O}_Y$-module $\mathcal{F}$ such that $\mathcal{O}_Y$ is $\pi$-typical to order $n$ and such that $\mathcal{F}$ is $\pi$-typical to order $n$ but not to order $n+1$, then there exists a proper morphism $f:X\to S$ such that $\mathcal{O}_X$ is $f$-typical to order $n$ but not to order $n+1$.
Proof. On $Y$, consider the direct sum of coherent sheaves, $$\mathcal{A}:= \mathcal{O}_Y\oplus \left( \mathcal{F}\cdot \epsilon \right),$$ where $\epsilon$ is a placeholder.  Since pushforward commutes with direct sums, this sheaf is $\pi$-typical to order $n$ but not to order $n+1$.  There is a unique structure of commutative, unital $\mathcal{O}_Y$-algebra on the $\mathcal{O}_Y$-module $\mathcal{A}$ such that $\epsilon \cdot \epsilon$ equals $0$.  This defines an $S$-scheme $X$ whose underlying topological space equals $Y$ and whose structure sheaf equals $\mathcal{A}$. QED
Rank $2$ vector bundles on the projective line.  The theory of rank $2$ vector bundles on the projective line is the same as the theory of Hirzebruch surfaces.  The basic phenomenon of these objects is that they can specialize and deform to one another.  We only need the simplest example of the Hirzebruch surface $\Sigma_0$ specializing to the Hirzebruch surface $\Sigma_2$.
Let $\rho:P\to S$ be a smooth, projective morphism whose geometric fibers are projective lines.  There is a universal extension on $P$ of the structure sheaf by the relative dualizing sheaf whose Yoneda Ext class, identified as an element of $H^1(P,\omega_{\rho})$, is a canonical generator with trace equal to $1$, $$0\to \omega_{\rho} \xrightarrow{\alpha} E_\rho \to \mathcal{O}_{P} \to 0.$$  For every choice of invertible sheaf $\mathcal{O}(1)$ on $P$ of $\rho$-relative degree $1$, there is a canonical isomorphism, $$E_\rho\cong \rho^*(\rho_*\mathcal{O}(1))\otimes \mathcal{O}(-1).$$  The extension class gives rise to a specialization of $E_\rho$ to the split extension, $\omega_{\rho}\oplus \mathcal{O}_{P}$.  More precisely, for every element $f\in \mathfrak{m}$, consider the cokernel $\mathcal{Q}_f$ on $P$ of the following morphism, $$(\alpha,f\cdot \text{Id}_{\omega}):\omega_{\rho}\to E\oplus \omega_{\rho}.$$ 
Lemma 3. The restriction of $\mathcal{Q}_f$ over the closed subscheme defined by $f$ equals the direct sum $\omega_{\rho}\oplus \mathcal{O}_{P}$.  The restriction of $\mathcal{Q}_f$ to the open complement of this closed subscheme equals $E_\rho$.  
Proof.  On the zero scheme of $f$, the restriction of $\mathcal{Q}_f$ equals $\mathcal{Q}_0$, which equals the cokernel of $\alpha$ direct sum with the structure sheaf.  On the other hand, on the open complement, $f$ is invertible.  Thus, the cokernel is the map to $E_\rho$ whose component on $\omega_{\rho}$ equals $-\alpha$.  QED
Our example $\mathcal{F}$ is a coherent sheaf on $Y=\mathbb{P}^2_S$ that is supported on a union of two lines.  On each line, the sheaf is of the form $\mathcal{Q}_f$ as above.  However, there is a novel glueing at the intersection point of the two lines.    
Let $n\geq 1$ be an integer.  On the zero scheme $\text{Zero}(t_2) \cong \mathbb{P}^1_S$, consider the cokernel $\mathcal{Q}'$ of the following homomorphism of locally free sheaves on $\mathbb{P}^2_S$, $$[t_0,t_1,s^n]^\dagger:\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2).$$  When restricted to $\text{Zero}(t_2,s^n)$, there is a short exact sequence, $$ 0 \to \mathcal{O}(-2) \to \mathcal{O}(-1)\oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2) \xrightarrow{[t_1,-t_0] \oplus \text{Id}} \mathcal{O}\oplus \mathcal{O}(-2) \to 0. $$
Similarly, on the zero scheme $\text{Zero}(t_1)\cong \mathbb{P}^1_k$, consider the cokernel $\mathcal{Q}''$ of the following homomorphism of locally free sheaves on $\mathbb{P}^2_S$, $$[t_2,t_0,s^n]^\dagger:\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2).$$  When restricted to $\text{Zero}(t_1,s^n)$, there is a short exact sequence, $$ 0 \to \mathcal{O}(-2) \to \mathcal{O}(-1)\oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2) \xrightarrow{[t_0,-t_2] \oplus \text{Id}} \mathcal{O}\oplus \mathcal{O}(-2) \to 0. $$
Note that at the common intersection point $\text{Zero}(s^n,t_1,t_2)$, the locally free quotients both restrict to free sheaves of rank $2$.  Choose an isomorphism of these free, rank $2$ sheaves so that the restrictions of the $\mathcal{O}$ summands do not match.  With respect to this isomorphism, there is a unique locally free sheaf $\mathcal{Q}$ of rank $2$ on $\text{Zero}(s^nt_2,t_1t_2)$ whose restriction to $\text{Zero}(t_2)$ equals $\mathcal{Q}'$, whose restriction to $\text{Zero}(s^n,t_1)$ equals $\mathcal{Q}''$, and such that the induced isomorphism between the restrictions to $\text{Zero}(s^n,t_1,t_2)$ of $\mathcal{Q}'$ and $\mathcal{Q}''$ is the given isomorphism.
Notation 4. Denote by $C_n$ the closed subscheme $C_n=\text{Zero}(t_1t_2,s^{n-1}t_2)$ of colength $1$.  Denote by $\mathcal{F}$ the restriction of $\mathcal{Q}$ to this closed subscheme $C_n$.  
Lemma 5. The pushforward $\pi_*\mathcal{F}$ is the zero sheaf.  
Proof. To see this, first consider the further restriction to $\text{Zero}(t_2)$.  This is a locally free sheaf of rank $2$ on $\text{Zero}(t_2) = \mathbb{P}^1_S$.  Thus, the pushforward to $S$ is a torsion-free coherent sheaf on $\mathbb{A}^1_S$, i.e., it is a free sheaf.  The restriction of $\mathcal{Q}$ to the generic fiber of $f$ equals $\mathcal{O}(-1)\oplus \mathcal{O}(-1)$ on $\mathbb{P}^1_{k(s)}$, and this has vanishing cohomology (it is an \emph{Ulrich sheaf}).  Thus, the pushforward of the restriction to $\text{Zero}(t_2)$ vanishes.
The kernel of the restriction to $\text{Zero}(t_2)$ is an $s^{n-1}$-torsion sheaf supported on $\text{Zero}(s^{n-1},t_1)$.  When restricted to $\text{Zero}(s,t_1)=\mathbb{P}^1_k$, this sheaf equals $\mathcal{O}(-1)\oplus \mathcal{O}(-3)$, the tensor product of $\mathcal{Q}''$ and the ideal sheaf of $\text{Zero}(t_1,t_2)$ inside $\text{Zero}(t_1)$.  Thus, this kernel has only the zero global section.  Altogether, $\pi_*\mathcal{F}$ is the zero sheaf. QED
Because of the lemma, $\mathcal{F}$ is $\pi$-typical to order $\ell$ if and only if $\pi_*\left(\mathcal{F}/\left( \mathfrak{m}^{\ell-1}\cdot \mathcal{F} \right) \right)$ is the zero sheaf.
Lemma 6. The sheaf $\mathcal{F}$ is $\pi$-typical to order $n$.
Proof. The restriction to each of $\text{Zero}(s^{n-1},t_2)$ and $\text{Zero}(s^{n-1},t_1)$ is isomorphic to $\mathcal{O}\oplus \mathcal{O}(-2)$. The isomorphism at the common intersection $\text{Zero}(s^{n-1},t_1,t_2)$ was chosen so that these global sections do not "glue".  Therefore $\mathcal{F}$ is $\pi$-typical of order $n$. QED
Proposition 7. The sheaf $\mathcal{F}$ is not $\pi$-typical to order $n+1$.
Proof. Consider the restrictions $\mathcal{F}_n$ and $\mathcal{F}_{n-1}$ of $\mathcal{F}$ to $\text{Zero}(s^n,s^{n-1}t_2,t_1t_2)$ and to $\text{Zero}(s^{n-1},s^{n-1}t_2,t_1t_2)$.  There is a natural surjection $\mathcal{F}_n\to \mathcal{F}_{n-1}$.  The kernel is the subsheaf of $\mathcal{Q}'$ equal to $s^{n-1}\cdot \mathcal{Q}'$.  The support of this sheaf equals $\text{Zero}(s,t_2)\cong \mathbb{P}^1_k$.  Since $\mathcal{Q}'$ equals $\mathcal{O}\oplus \mathcal{O}(-2)$ on $\text{Zero}(s^n,t_2)$, this kernel sheaf equals $\mathcal{O}\oplus \mathcal{O}(-1)$ on $\mathbb{P}^1_k$.  
Thus, the kernel sheaf has a $1$-dimensional $k$-vector space of global sections.  Since this kernel sheaf is a subsheaf of $\mathcal{F}/\left( s^n\cdot \mathcal{F} \right)$, also this sheaf has a nonzero $k$-vector space of global sections.  Since the domain of the map $r_n$ is the zero vector space, it follows that $r_n$ is not surjective.  Therefore the sheaf $\mathcal{F}$ is $\pi$-typical to order $n$, but it is not $\pi$-typical to order $n+1$.  QED
