Cherednik algebras and Hilbert schemes in characteristic $p$

We prove a localization theorem for the type $\mathbf{A}_{n-1}$ rational Cherednik algebra $\mathsf{H}_c=\mathsf{H}_{1,c}(\mathbf{A}_{n-1})$ over $\overline{\mathbb{F}}_p$, an algebraic closure of the finite field. In the most interesting special case where $c\in \mathbb{F}_p$, we construct an Azumaya algebra $\mathscr{H}_c$ on $\operatorname{Hilb}^n{\mathbb{A}}^2$, the Hilbert scheme of $n$ points in the plane, such that $\Gamma(\operatorname{Hilb}^n{\mathbb{A}}^2, \,\mathscr{H}_c)=\mathsf{H}_c$. Our localization theorem provides an equivalence between the bounded derived categories of $\mathsf{H}_c$-modules and sheaves of coherent $\mathscr{H}_c$-modules on $\operatorname{Hilb}^n{\mathbb{A}}^2$, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.