Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

TL;DR

This paper explores how a modified derived McKay correspondence simplifies the analysis of tautological bundles on Hilbert schemes of points, leading to new formulas and shorter proofs for homological invariants.

Contribution

It introduces a slightly different derived equivalence that simplifies the study of tautological bundles and provides new formulas for their homological invariants.

Findings

Simplified proofs for known results

New formulas for homological invariants

Computed extension groups between wedge powers

Abstract

We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.