On Localization for Quantum Hamiltonian Reductions in Arbitrary Characteristic

TL;DR

This paper characterizes when derived localization holds for quantum Hamiltonian reductions in any characteristic, linking it to finite global dimension and Morita equivalences, with applications to hypertoric varieties in positive characteristic.

Contribution

It provides an explicit criterion for derived localization based on a finite set of line bundles, extending understanding in arbitrary characteristic.

Findings

Derived localization holds if and only if it holds for specific line bundles.

Localization is proven for certain weights in hypertoric cases when p exceeds a bound.

Derived localization follows from a finite number of Morita equivalences.

Abstract

For quantum Hamiltonian reductions in arbitrary characteristics, it is known that derived localization holds if and only if the algebra of global sections has finite global dimension. In this paper we provide an alternative characterization of when derived localization holds: Derived localization holds if and only if it holds for an explicit finite set of (quantized) line bundles. As an application, we prove a new result that there are integral weights for which localization holds on in the positive characteristic hypertoric case for $p$ larger than an explicit bound. We also discuss how derived localization is a consequence of a finite number of Morita equivalences.