Kernels from Compactifications

TL;DR

This paper introduces a homotopical method to construct kernels inducing semi-orthogonal decompositions in equivariant derived categories, providing new proofs of wall-crossing and extending to $D$-flips.

Contribution

It develops a novel homotopical construction of kernels from affine schemes with $ ext{G}_m$-actions, leading to semi-orthogonal decompositions and new insights into wall-crossing phenomena.

Findings

Constructed idempotent integral kernels for affine schemes with $ ext{G}_m$-actions.

Provided a new proof of wall-crossing equivalences using these kernels.

Extended the construction to integral transforms related to $D$-flips.

Abstract

To any affine scheme with a $\mathbb{G}_m$-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant derived category with a canonical semi-orthogonal decomposition. As a special case, we demonstrate that grade-restriction windows appear as a consequence of this construction, giving a new proof of wall-crossing equivalences which works over an arbitrary base. The construction globalizes to yield interesting integral transforms associated to $D$-flips.