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Categorical characterizations of regularity for algebraic stacks

This paper extends Neeman's categorical characterizations of regularity from Noetherian schemes to Noetherian algebraic stacks by proving that regularity is equivalent to the equality of perfect and bounded coherent derived categories, thereby establishing the existence of strong generators and bounded $t$ -structures for these stacks.

Original authors: Timothy De Deyn, Pat Lank, Kabeer Manali Rahul, Fei Peng

Published 2026-04-21

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CC BY 4.0

Original authors: Timothy De Deyn, Pat Lank, Kabeer Manali Rahul, Fei Peng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the structural integrity of a building. In the world of mathematics, specifically algebraic geometry, the "buildings" are shapes called schemes and stacks. Some of these buildings are perfectly smooth and sturdy (called regular), while others have cracks, jagged edges, or hidden weaknesses (called singular).

For a long time, mathematicians knew how to tell if a simple building (a scheme) was sturdy just by looking at its internal "blueprints" (mathematical objects called derived categories). They found two special keys:

Strong Generators: A single blueprint that can be used to build every other blueprint in the set using a limited number of steps.
Bounded T-structures: A way of organizing the blueprints into neat, finite layers, like a well-organized library.

If you could find these keys, the building was guaranteed to be "regular" (sturdy). If you couldn't, it had flaws.

The Problem:
The authors of this paper, Timothy De Deyn and his team, wanted to apply these rules to much more complex structures called algebraic stacks. Think of a stack not as a simple building, but as a building with a twist: it has "ghosts" or "symmetries" inside it. For example, a point in a stack might actually represent a whole group of identical points rotating around each other. These symmetries make the math messy. In these complex buildings, the old blueprints (perfect complexes) don't always match the compact objects (the building blocks), so the old rules sometimes break.

The Solution:
The team successfully adapted the rules to work for these complex, symmetrical buildings. Here is what they discovered, explained simply:

1. The "Perfect Match" Rule (Proposition 1.3)

In a simple, sturdy building, the set of "perfect blueprints" (objects that behave nicely) is exactly the same as the set of "bounded coherent blueprints" (objects that are finite and manageable).

The Metaphor: Imagine a library where every book is a perfect, self-contained story. If the library is "regular," then every book in the "bounded" section (the short stories) is also a "perfect" book.
The Discovery: The authors proved that for these complex stacks, if the "perfect blueprints" and the "bounded blueprints" are the same set, the building is regular. If they are different, the building has a crack.

2. The "Master Key" Rule (Theorem 1.1)

This is about Strong Generators.

The Metaphor: Imagine you have a giant box of LEGO bricks. If the building is regular, there is one specific "Master Brick" (a strong generator) that you can use, along with a few simple moves (stacking, splitting, shifting), to build any other structure in the box.
The Discovery: The authors proved that for a wide class of these complex stacks, if you can find this one "Master Brick" that can generate the whole system, the stack is regular. If you can't find such a brick, the stack is irregular.

3. The "Organized Library" Rule (Theorem 1.2)

This is about Bounded T-structures.

The Metaphor: Imagine a chaotic pile of papers. A "bounded t-structure" is a way of sorting them into neat, finite folders (layers) so that no paper is lost in an infinite abyss.
The Discovery: They showed that if you can organize the "perfect blueprints" of a specific part of the stack into these neat, finite folders, that specific part of the stack is regular. If the papers are hopelessly jumbled and you can't sort them into finite layers, that part of the stack is broken.

Why This Matters

Before this paper, these powerful tools only worked for simple buildings (schemes). The authors had to invent new ways to handle the "ghosts" and symmetries of stacks. They had to prove that even with these symmetries, the fundamental relationship between the building's geometry and its mathematical blueprints holds true.

In a Nutshell:
The paper says: "We have found that even for the most complex, symmetrical mathematical buildings, you can still tell if they are structurally sound by checking if their internal blueprints can be generated by a single master piece or organized into neat, finite layers. If they can, the building is perfect. If not, it's broken."

This is a big deal because it allows mathematicians to use these powerful, high-level tools to study complex geometric objects that were previously too difficult to analyze.

1. Problem Statement

The paper addresses the extension of categorical characterizations of regularity from Noetherian schemes to Noetherian algebraic stacks.

In the context of schemes, Neeman (2021, 2023) established that for a Noetherian scheme $X$ of finite Krull dimension:

$X$ is regular if and only if the category of perfect complexes, $\text{Perf}(X)$ , admits a strong generator.
$X$ is regular if and only if $\text{Perf}(X)$ admits a bounded $t$ -structure.

The authors aim to generalize these results to algebraic stacks. A significant hurdle is that for general algebraic stacks, the category of perfect complexes ( $\text{Perf}(X)$ ) does not necessarily coincide with the subcategory of compact objects in the unbounded derived category of quasi-coherent sheaves ( $D_{qc}(X)^c$ ). This discrepancy complicates the direct application of Neeman's scheme-theoretic proofs, which often rely on compact generation properties.

2. Methodology and Technical Framework

The authors employ a combination of derived category theory, stack geometry, and inductive principles to overcome the obstacles posed by the stacky setting.

Concentrated Stacks: The authors focus on concentrated algebraic stacks (where $\text{Perf}(X) = D_{qc}(X)^c$ ). This assumption is crucial because it restores the equivalence between perfect complexes and compact objects, a property that holds for schemes but fails for general stacks (e.g., classifying stacks of positive characteristic groups).
Homological Characterization of Regularity: A central intermediate result is establishing that for decent, quasi-compact, locally Noetherian stacks, regularity is equivalent to the equality $\text{Perf}(X) = D^b_{\text{coh}}(X)$ . This bridges the gap between the geometric definition of regularity and the categorical properties of derived categories.
Induction Principles: The proofs utilize the induction principle for algebraic stacks (Hall–Rydh), allowing the authors to reduce global properties to local checks on open substacks and closed substacks via recollements.
Recollements and Gluing: The paper heavily relies on the theory of recollements of triangulated categories. The authors analyze how $t$ -structures and generators behave under gluing along closed immersions and open immersions.
Hypothesis $\star$ : To handle $t$ -structures, the authors introduce a technical condition called Hypothesis $\star$ . A stack satisfies this if it is 1-Thomason and the standard $t$ -structure on the derived category of complexes supported on a closed subset lies in the "preferred equivalence class" (generated by compact objects). They prove this holds for stacks with quasi-finite separated diagonals and Deligne–Mumford stacks of characteristic zero.

3. Key Contributions and Results

A. Equivalence of Regularity and Perfect/Coherent Equality

Proposition 1.3 / Corollary 3.8: For a decent, quasi-compact, locally Noetherian algebraic stack $X$ , $X$ is regular if and only if $\text{Perf}(X) = D^b_{\text{coh}}(X)$ .

Significance: This extends a classical scheme-theoretic fact to stacks. The proof involves detecting regularity at closed points via the perfectness of structure sheaves of residual gerbes, circumventing the need for compact objects in non-concentrated cases.

B. Strong Generators and Regularity

Theorem 1.1 / Theorem 5.4: Let $X$ be a concentrated, separated, Noetherian algebraic stack of finite Krull dimension satisfying "approximation by compacts." Then $X$ is regular if and only if $\text{Perf}(X)$ admits a strong generator.

Context: This generalizes Neeman's result [Nee21, Theorem 0.5]. The "approximation by compacts" condition is satisfied by stacks with quasi-finite separated diagonals.
Note: The existence of a strong generator implies finite Krull dimension in the Noetherian case, so this dimension constraint cannot be relaxed.

C. Bounded $t$ -Structures and Regularity

Theorem 1.2 / Theorem 6.7: Let $X$ be a concentrated Noetherian algebraic stack of finite Krull dimension satisfying Hypothesis $\star$ (e.g., quasi-finite separated diagonal or Deligne–Mumford of char 0). For any closed subset $Z \subseteq |X|$ , $Z \subseteq \text{reg}(X)$ if and only if $\text{Perf}_Z(X)$ admits a bounded $t$ -structure.

Significance: This generalizes Neeman's relative result [Nee23a]. The proof avoids establishing "weak approximability" for $D_{qc,Z}(X)$ , which is a difficult technical step in the scheme case, by instead proving that the standard $t$ -structure lies in the preferred equivalence class.

D. Classical Generators for Coherent Sheaves

Proposition 1.5 / Proposition 4.6: The paper provides criteria for the existence of classical generators for $D^b_{\text{coh}}(Z)$ for integral closed substacks.

Specifically, if $X$ is concentrated and satisfies the 1-Thomason condition, $D^b_{\text{coh}}(Z)$ admits a classical generator if and only if the regular locus $\text{reg}(Z)$ is open (or contains a nonempty open for every integral substack).
This generalizes results by Iyengar–Takahashi and Dey–Lank from commutative rings and schemes to stacks.

4. Significance and Impact

Unification of Geometry and Category Theory: The paper successfully translates deep homological properties of regularity (previously known for schemes) into the language of algebraic stacks, confirming that the categorical behavior of derived categories remains a robust invariant of geometric regularity even in the presence of stabilizers.
Handling Non-Concentrated Stacks: By explicitly addressing the failure of $\text{Perf}(X) = D_{qc}(X)^c$ in general stacks (e.g., $B\mathbb{G}_a$ in characteristic $p$ ), the authors clarify the precise conditions (concentratedness) under which Neeman's theorems hold.
New Technical Tools: The development of Hypothesis $\star$ and the verification of its validity for quasi-finite and Deligne–Mumford stacks provides a new framework for studying $t$ -structures on stacks. This avoids the need for weak approximability, offering a potentially more accessible path for future research in derived algebraic geometry.
Generators for Coherent Categories: The results on classical generators for $D^b_{\text{coh}}$ extend the understanding of when bounded derived categories of coherent sheaves are "finite" in a categorical sense, which is relevant for non-commutative geometry and the study of singularities.

In summary, this work bridges a critical gap in derived algebraic geometry, proving that the powerful categorical characterizations of regularity developed for schemes are robust enough to apply to a wide and important class of algebraic stacks, provided specific finiteness and concentration conditions are met.