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Categorical absorption for hereditary orders

Thilo Baumann

TL;DR

This work develops a noncommutative analogue of deformation absorption of singularities for hereditary orders on curves. By treating a hereditary $\mathcal{O}_C$-order with ramification at a point as a smoothing over a base curve and analyzing its ramified fiber via $\Lambda_r$-type simple modules, the authors construct a $C$-linear semiorthogonal decomposition of $\mathsf{D}^{\text{b}}(C,\mathscr{A})$ that isolates a smooth, base-proper component equivalent to $\mathsf{D}^{\text{b}}(C)$. The decomposition is organized by a sequence of $\mathbb{P}^{\infty,2}$-objects whose pushforwards induce exceptional pieces, yielding a deformation-absorption structure and a $2r$-periodicity under Serre duality. The paper also connects the noncommutative picture to smooth root-stack dictionaries, providing a coherent stacks--orders correspondence and extending noncommutative base-change results to $\mathsf{D}^{\text{b}}(X,\mathscr{A})$. Overall, it offers a new, base-linear framework for understanding the derived categories of hereditary orders through singularity absorption and root-stack geometry.

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.

Categorical absorption for hereditary orders

TL;DR

This work develops a noncommutative analogue of deformation absorption of singularities for hereditary orders on curves. By treating a hereditary -order with ramification at a point as a smoothing over a base curve and analyzing its ramified fiber via -type simple modules, the authors construct a -linear semiorthogonal decomposition of that isolates a smooth, base-proper component equivalent to . The decomposition is organized by a sequence of -objects whose pushforwards induce exceptional pieces, yielding a deformation-absorption structure and a -periodicity under Serre duality. The paper also connects the noncommutative picture to smooth root-stack dictionaries, providing a coherent stacks--orders correspondence and extending noncommutative base-change results to . Overall, it offers a new, base-linear framework for understanding the derived categories of hereditary orders through singularity absorption and root-stack geometry.

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.

Paper Structure

This paper contains 27 sections, 35 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Categorical absorption and deformation absorption of singularities.
  3. Noncommutative base change.
  4. Organization of the paper.
  5. Notations and conventions.
  6. Acknowledgements.
  7. Coherent ringed schemes
  8. Modules on coherent ringed schemes.
  9. Fiber products of coherent ringed schemes.
  10. K-injectives and locally projectives.
  11. The derived category of a coherent ringed scheme.
  12. Semiorthogonal decompositions linear over the base.
  13. Hereditary orders
  14. The fibers of hereditary orders.
  15. Maximal overorders.
  16. ...and 12 more sections

Key Result

Theorem 1

The sequence $(S_1,\ldots, S_{r-1})$ of simple $\mathscr{A}(o)$-modules is semiorthogonal in $\mathsf{D}^{\mathsf{b}}(\mathscr{A}(o))$ and absorbs singularities, i.e. the triangulated subcategory is admissible and both of its complements ${}^\perp \mathsf{S}$ and $\mathsf{S}^\perp$ are smooth and proper.

Theorems & Definitions (80)

  • Theorem 1: \ref{['thm:absorption-of-singularities']}
  • Theorem 2: \ref{['thm:deformation-absorption']}
  • Theorem 3: \ref{['thm:main-base-change-theorem']}
  • definition 2.1
  • remark 2.2
  • remark 2.3
  • lemma 2.4
  • proof
  • remark 2.5
  • definition 2.6
  • ...and 70 more