Categorical absorption for hereditary orders

TL;DR

This paper introduces a new method for analyzing the derived categories of hereditary orders on curves by leveraging deformation absorption of singularities, leading to semiorthogonal decompositions.

Contribution

It develops a novel approach using deformation absorption to study derived categories of hereditary orders, connecting singularity theory with categorical decompositions.

Findings

Constructed a triangulated subcategory for finite-dimensional algebra

Provided a semiorthogonal decomposition of the derived category

Demonstrated linearity over the base for the decomposition

Abstract

We show that Kuznetsov--Shinder's notion of deformation absorption of singularities leads to a new approach for studying the bounded derived category of a hereditary order on a curve. The starting point is a hereditary order which can be interpreted as a smoothing of the finite-dimensional algebra obtained from the restriction to a ramified point. We construct a triangulated subcategory inside the derived category of this finite-dimensional algebra which provides a deformation absorption of singularities. This allows us to obtain a semiorthogonal decomposition of the bounded derived category of the hereditary order, which is in addition linear over the base.