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Nonexistence of singly compactly generated $t$-structures for schemes

Anirban Bhaduri, Timothy De Deyn, Michal Hrbek, Pat Lank, Kabeer Manali-Rahul

TL;DR

This work proves that the standard aisle $D^{\leq 0}_{qc}(\mathcal{X})$ on derived categories of quasi-coherent sheaves cannot be singly compactly generated in new classes of schemes and stacks, including $\mathbb{P}^1_k$ and proper tame DM stacks of positive dimension. It develops a framework linking compact generation of aisles to weak generators of $\operatorname{coh}(\mathcal{X})$ and shows that, in locally Noetherian settings with approximation by compacts, a singly generated aisle forces $\operatorname{coh}(\mathcal{X})$ to be Artinian; in turn, for proper spaces over a field this implies the space is affine Artinian. The results extend to general bases via base-change and yield that proper morphisms with positive relative dimension cannot have singly generated standard aisles, providing arithmetic-type counterexamples and preventing broad classes of schemes from admitting such t-structures. Overall, the paper reveals structural obstructions to compact generation of standard t-structure aisles and identifies precise geometric consequences (Artinian-ness, finiteness) for the underlying spaces.

Abstract

We show the first instances of schemes whose standard aisles on their derived category of quasi-coherent sheaves are not singly compactly generated.

Nonexistence of singly compactly generated $t$-structures for schemes

TL;DR

This work proves that the standard aisle on derived categories of quasi-coherent sheaves cannot be singly compactly generated in new classes of schemes and stacks, including and proper tame DM stacks of positive dimension. It develops a framework linking compact generation of aisles to weak generators of and shows that, in locally Noetherian settings with approximation by compacts, a singly generated aisle forces to be Artinian; in turn, for proper spaces over a field this implies the space is affine Artinian. The results extend to general bases via base-change and yield that proper morphisms with positive relative dimension cannot have singly generated standard aisles, providing arithmetic-type counterexamples and preventing broad classes of schemes from admitting such t-structures. Overall, the paper reveals structural obstructions to compact generation of standard t-structure aisles and identifies precise geometric consequences (Artinian-ness, finiteness) for the underlying spaces.

Abstract

We show the first instances of schemes whose standard aisles on their derived category of quasi-coherent sheaves are not singly compactly generated.

Paper Structure

This paper contains 10 sections, 10 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abelian categories
  4. t-structures
  5. Algebraic stacks
  6. Results
  7. Locally Noetherian Abelian categories
  8. Proper over a field
  9. Proper over general base
  10. An alternative approach: Stability

Key Result

Theorem 1

This is false for $\mathbb{P}^1_k$ over a field $k$. In fact, it fails more generally for proper tame Deligne--Mumford stacks $\mathcal{X}$ of positive Krull dimension over $k$.

Theorems & Definitions (22)

  • Theorem : see \ref{['thm:coarse_moduli']}
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • ...and 12 more