## 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 $$0) and is not an isomorphism at any later stage?
• Is there a description of $$(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 $$X$$?