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$ .

  • 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?

Flatness properties of Poincare sheaf on relative compactified Jacobians

Assume we are given a $ K3$ -surface $ X$ with $ Pic(X)=\mathbb{Z}\mathcal{O}_X(1)$ such that $ \mathcal{O}_X(1)$ ample and of degree 2.

Then we have the linear system $ |\mathcal{O}_X(1)|=\mathbb{P}^2$ and a general member is a smooth curve of genus 2.

Let $ f: C\rightarrow |\mathcal{O}_X(1)|$ be the universal curve, that is $ C=\{(x,D)\,|\, x\in D\}\subset X\times |\mathcal{O}_X(1)|$ .

Denote by $ \overline{J}^d$ the relative compactified Jacobian of degree $ d$ of the family $ f: C\rightarrow |\mathcal{O}_X(1)|$ .

If $ d$ is even, then $ \overline{J}^d$ is a fine moduli space and hence there is a Poincare sheaf $ U$ over $ C\times_{|\mathcal{O}_X(1)|}\overline{J}^d$ flat over $ \overline{J}^d$ .

$ \textbf{Question:}$ Is $ U$ flat over $ C$ ?

We also have $ C\times_{|\mathcal{O}_X(1)|}\overline{J}^d\subset X\times \overline{J}^d$ . Is $ U$ seen as a sheaf on $ X\times \overline{J}^d$ flat over $ X$ ?