Linking regularity of ideal sheaf with Fitting ideals sheaf

Question

I'm reading Eisenbud's book The geometry of syzygies and I'm quite struck undestanding the argument proposed in Chapter 5, in the section named "Fitting ideals". Remember that a coherent sheaf $\mathscr{F}$ over a scheme $X$ is called $m$*-regular* if $$H^p(X,\mathscr{F}(m-p))=0$$ for every $p>0$. The minimum $m$ among integers, if it exists, such that $\mathscr{}$ is $m$-regolar is the Castelnuovo-Mumford regularity $\mathrm{reg}(\mathscr{F})$. There is an algebraic definition for module that involves (among other things) local cohomology.

Let be $X$ an irreducible and smooth projective curve over an algebraically closed field $k$. If we call $I(X)$ the homogeneous saturated ideal of $X$, id est

$$ I(X):=\bigoplus_{n\geq 0} \mathscr{I}_X(n) $$

where $\mathscr{I}_X$ is the ideal sheaf of $X$, we know thath $\mathrm{reg}(I(X))=\mathrm{reg}(\mathscr{I}_X)$.

Now, let suppose that $X$ comes with a very ample line bundle, i.e. lets assume that $X\subseteq \mathbf{P}^r_k$ for some $r\geq 2$; in this case we have $I(X)\subseteq k[x_0,\ldots,x_r]=:S$ as homogeneous ideal. Let be $\mathscr{L}\in\mathrm{Pic}(X)$, i.e. an invertibile sheaf over $X$ and let be $$F:=\bigoplus_{n\geq 0} H^0(X,\mathscr{L}(n))$$ the generated cone; it's well known that $F$ is a graded $S$-module and has a free minimal presentation $$L_1\overset{\psi}{\longrightarrow} L_0\longrightarrow F\longrightarrow 0$$

Sheafifying this sequence, we obtain a sequence

$$\bigoplus_{j=1}^s \mathscr{O}_{\mathbf{P}^r_k}(-h_j)\overset{\Psi}{\longrightarrow} \bigoplus_{l=1}^t \mathscr{O}_{\mathbf{P}^r_k}(-l_i)\longrightarrow \mathscr{L}\longrightarrow 0$$

In both cases we can define

• the 0-th Fitting ideal $I(\psi)$ of $\psi$, that is the ideal generated by $\psi$'s maximal minors;

• the sheaf $\mathscr{I}(\Psi)$ of Fitting ideals of $\Psi$.

The problems that arise are the following:

1. how can we relate the sheaf $\widetilde{I(\psi)}$ to the sheaf $\mathscr{I}(\psi)$? Since Fitting ideals commute with localizations, I find quite reasonable that the two sheafs are isomorphic, but Eisenbud don't even give an hint of a formal way to prove it;

2. What can we say about $\mathrm{reg}(\mathscr{I}(\Psi))$ and $\mathrm{reg}(\mathscr{I}_X)=\mathrm{reg}(I(X))$? Following Eisenbud considerations (which don't assume $X$ smooth), there should be inequality $\mathrm{reg}(\mathscr{I}_X)\leq\mathrm{reg}(\mathscr{I}(\Psi))$, but following the cumbersome proof of this fact it seems that, if $X$ is smooth, then we have $\mathscr{I}_X=\mathscr{I}(\Psi)$.

Anyone can help me finding the way to cleat out these facts?