Categorical absorption for hereditary orders

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.