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Measuring birational derived splinters

Timothy De Deyn, Pat Lank, Kabeer Manali-Rahul, Sridhar Venkatesh

TL;DR

This work develops a derived-category approach to birational singularities by introducing the invariant $_{ ext{bds}}(X)$, which quantifies the failure of a Noetherian scheme to be a birational derived splinter via levels in $R f_st D^b_{oh}(Y)$. The authors connect this invariant to the conjectural landscape around splinters, rational singularities, and derived splinters across characteristics, and show that, under resolutions, the invariant is finite and computable via blowups. They establish fundamental base-change, pullback, and local-to-global principles for $_{ ext{bds}}$, including stability under smooth pullbacks, behavior under completions, and equalities for projective-bundle constructions. The paper then derives consequences for descent and ascent of birational derived splinters under field extensions and products, providing sharp inequalities that clarify how birational-derived-splinter properties interact with base change and fiber products. Overall, the results advance a categorical toolkit for understanding singularities in all characteristics and lay groundwork for further exploration of birational derived splinters via generation in triangulated categories.

Abstract

This work is concerned with categorical methods for studying singularities. Our focus is on birational derived splinters, which is a notion that extends the definition of rational singularities beyond varieties over fields of characteristic zero. Particularly, we show that an invariant called `level' in the associated derived category measures the failure of these singularities.

Measuring birational derived splinters

TL;DR

This work develops a derived-category approach to birational singularities by introducing the invariant , which quantifies the failure of a Noetherian scheme to be a birational derived splinter via levels in . The authors connect this invariant to the conjectural landscape around splinters, rational singularities, and derived splinters across characteristics, and show that, under resolutions, the invariant is finite and computable via blowups. They establish fundamental base-change, pullback, and local-to-global principles for , including stability under smooth pullbacks, behavior under completions, and equalities for projective-bundle constructions. The paper then derives consequences for descent and ascent of birational derived splinters under field extensions and products, providing sharp inequalities that clarify how birational-derived-splinter properties interact with base change and fiber products. Overall, the results advance a categorical toolkit for understanding singularities in all characteristics and lay groundwork for further exploration of birational derived splinters via generation in triangulated categories.

Abstract

This work is concerned with categorical methods for studying singularities. Our focus is on birational derived splinters, which is a notion that extends the definition of rational singularities beyond varieties over fields of characteristic zero. Particularly, we show that an invariant called `level' in the associated derived category measures the failure of these singularities.

Paper Structure

This paper contains 12 sections, 23 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Measurements
  3. Some consequences
  4. Preliminaries
  5. Generation
  6. Types of morphisms
  7. Birational derived splinters
  8. Measurements
  9. First steps
  10. Behavior under pullback
  11. Local behavior
  12. Some consequences

Key Result

Lemma 1.2

Let $f\colon Y \to X$ be proper birational (more generally surjective) morphism to a Noetherian scheme. Then the natural morphism $\mathcal{O}_X \to \mathbf{R}f_\ast \mathcal{O}_Y$ splits if, and only if, $D^b_{\operatorname{coh}}(X) = \langle \mathbf{R}f_\ast D^b_{\operatorname{coh}}(Y) \rangle_1$.

Theorems & Definitions (50)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Theorem 1.4: see \ref{['prop:independence_field', 'prop:submultiplicativity']}
  • Corollary 1.5: see \ref{['thm:base_change_bds_field']} and \ref{['cor:fibered_product_bds']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 40 more