odule by Lemma \[hi-basic-lem\]. By induction on $m$, it follows that $\operatorname{{\textsf}{ord}}^mN(i)$ is also free. The analogous results for $B_{ij}$ follow by multiplying everything on the right by $e$.

 {#cohh-subsect}

We end the section by noting that Proposition \[pre-cohh\] provides an interesting connection between $H_c$-modules and the isospectral scheme $X_n$ defined in . Adjusting to the conventions of this section, we identify $\operatorname{Hilb(n)}= \operatorname{Proj}\widetilde{A}$, for $\widetilde{A}=\bigoplus A^k\delta^k$. By construction, the Procesi bundle ${\mathcal{P}}=\rho_*{\mathcal{O}}_{X_n}$ from is then just the image in $\operatorname{{\textsf}{Coh} }\, \operatorname{Hilb(n)}$ of the $\widetilde{A}$-module $\bigoplus J^k\delta^k$. Thus the next result is an immediate consequence of Proposition \[pre-cohh\].

Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\]. Let $e\widetilde{H}_c= \bigoplus_{k\geq 0}B_{k0}\otimes_{U_c} e\widetilde{H}_c$ be the $B$-module associated to the $U_c$-module $eH_c$ and filter each $B_{k0}\otimes_{U_c}eH_c \cong B_{k0}eH_c$ by the $\operatorname{{\textsf}{ord}}$ filtration. Set $\operatorname{{\textsf}{ogr}}e\widetilde{H}_c = \bigoplus \operatorname{{\textsf}{ogr}}B_{k0}eH_c$. Then the sheaf associated to $\operatorname{{\textsf}{ogr}}e\widetilde{H}_c$ in $\operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$ is the Procesi bundle ${\mathcal{P}}$.

 {#cohh-subsect-chat}

Just as Theorem \[main\] can be interpreted as saying that $U_c$ provides a noncommutative model for $\operatorname{Hilb(n)}$, so Corollary \[cohh-subsect\] can be interpreted as saying that the algebra $H_c$ provides a noncommutative model for $X_n$. Here is one aspect of this analogy. It follows from [@BKR] and [@hai1] that there is an equivalence $\xi$ of derived categories between ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and $\operatorname{Hilb(n)}$ that is induced by a Fourier-Mukai transform over ${\mathcal{P}}$. Now pass to the noncommutative situation, 