Nilpotent cohomological Hall algebras of surfaces

This paper establishes a framework for cohomological Hall algebras associated with coherent sheaves supported on a fixed curve within a smooth surface, constructing a generalized moduli stack and defining a functorial algebra that depends only on the formal neighborhood of the curve to study Hecke operators and resolve questions regarding Kleinian singularities.

The Gist

Imagine you are looking at a smooth, flat sheet of fabric (a mathematical "surface"). Now, imagine drawing a specific line or shape on that fabric. This shape might be a simple circle, or it might be a messy, tangled knot where the fabric folds over itself (a "singular" or "reducible" curve).

This paper is about building a new kind of mathematical machine (an algebra) that helps us understand how we can tweak or "modify" the fabric specifically along that drawn line, without worrying about what's happening far away from it.

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The Problem: Too Many Possibilities

In mathematics, when you study how to change a fabric along a line, you usually have to look at the whole fabric. But sometimes, the changes you care about are so specific to that line that the "whole fabric" view is too messy and infinite. It's like trying to understand how a specific thread in a sweater is knotted by looking at the entire ocean.

The authors wanted to create a system that focuses only on the neighborhood of that specific line, ignoring the rest of the universe. They call this the "formal neighborhood."

2. The Solution: A "Zoom-In" Machine

The paper constructs a new mathematical object called a Nilpotent Cohomological Hall Algebra (COHA).

• The "Hall" part: Think of this as a rulebook for combining things. If you have two different ways of modifying the fabric along the line, this rulebook tells you how to "multiply" them to get a third way.
• The "Nilpotent" part: This is the key filter. It means the machine only cares about modifications that are "zero" or "trivial" if you move them too far away from the line. It's like a spotlight that only illuminates the line itself; anything outside the light fades into nothingness.
• The "Cohomological" part: This is the measuring tape. It doesn't just count the modifications; it measures their "shape" and "twists" using advanced geometry.

3. The Big Discovery: The "Local" Secret

The most important finding in the paper is that this new machine only depends on the immediate neighborhood of the line, not the whole surface.

• The Analogy: Imagine you have a map of the world. Usually, to understand a specific city, you need to know the whole country. This paper proves that for these specific types of fabric modifications, you can tear the map out, keep only the tiny square inch containing the city, and you will get the exact same mathematical results.
• Why it matters: This allows mathematicians to do "local" calculations (which are easier) and know they apply to the "global" situation. It turns a massive, impossible puzzle into a small, manageable one.

4. The "Moduli Stack": A Catalog of All Possibilities

To build this machine, the authors first had to create a giant catalog (called a "moduli stack") of every possible way to modify the fabric along that line.

• They proved that even though this catalog is infinitely large, it has a very organized structure. It's like a library that is infinitely tall, but if you look at the "reduced" version (stripping away the complex, fuzzy details), it looks like a standard, well-organized building.
• This structure allows them to define the "Borel-Moore homology," which is essentially a way to count and measure the "holes" and "loops" in this infinite library.

5. The Connection to Other Math

The paper mentions that this new machine connects to other famous mathematical tools:

• Hecke Operators: These are like "switches" that change the state of the fabric. The authors show their new machine is the "biggest possible switchboard" for these changes along the line.
• Quantum Groups and Yangians: These are complex algebraic structures used in physics (like quantum mechanics). The paper sets the stage for showing how these fabric-modification machines are actually the same as those physics machines, specifically when the fabric is a "minimal resolution" of a singularity (a way of smoothing out a sharp point).

Summary

In simple terms, this paper builds a specialized calculator for studying how to tweak a surface along a specific, possibly messy line.

1. It proves that you can study this line in isolation (locally) without needing to know the whole surface.
2. It creates a rulebook (an algebra) for combining these tweaks.
3. It shows that this rulebook is robust and works whether you are looking at the whole surface or just the tiny neighborhood of the line.

This work doesn't just solve a puzzle; it provides the foundation (the "framework") for other mathematicians to use these tools to solve even harder problems, such as connecting geometry to quantum physics, which the authors mention they do in a companion paper.

Technical Summary

Technical Summary: Nilpotent Cohomological Hall Algebras of Surfaces

Problem and Motivation
The paper addresses the systematic study of cohomological "Hecke operators" associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (which may be singular and reducible). While Cohomological Hall Algebras (COHAs) for smooth surfaces and preprojective algebras of quivers are well-established, a general framework for "nilpotent" COHAs—specifically those governing sheaves set-theoretically supported on a subscheme $Z$—was lacking.

The authors aim to construct a moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, denoted $\text{Coh}(\widehat{X}_Z)$, and endow its Borel-Moore homology with a canonical Hall algebra structure. This construction is motivated by the need to understand the relationship between the COHA of a minimal resolution of a Kleinian singularity and the corresponding preprojective COHA (a question addressed in the companion paper [DPS$^+$26]) and to provide an intrinsic construction of the "largest" algebra of cohomological Hecke operators for such modifications.

Methodology and Framework
The authors develop a sophisticated framework combining derived algebraic geometry, motivic homotopy theory, and the theory of ind-objects.

1. Moduli Stacks and Ind-Geometric Structures:
   The central object is the moduli stack $\text{Coh}(\widehat{X}_Z)$ of coherent sheaves on the formal completion $\widehat{X}_Z$ of $X$ along $Z$. Since this category is not of finite type, the stack is not an algebraic stack in the classical sense. The authors prove that $\text{Coh}(\widehat{X}_Z)$ is an indgeometric stack (a filtered colimit of Artin stacks with closed immersion transition maps).

   • Theorem A: They establish that the reduced stack $\text{Coh}(\widehat{X}_Z)_{\text{red}}$ is a quasi-separated Artin stack locally of finite type. Furthermore, $\text{Coh}(\widehat{X}_Z)$ is identified as the formal completion of the stack of all coherent sheaves $\text{Coh}_{\text{ps}}(X)$ along the closed substack of sheaves supported on $Z$.
   • Admissibility: To handle the non-quasi-compactness, the authors introduce the class of admissible indgeometric stacks, characterized by having a geometric reduced stack. They construct an explicit admissible open exhaustion for these stacks using Harder-Narasimhan strata.

2. Motivic Borel-Moore Homology:
   Extending the work of A. Khan [Kha19] and the authors' previous work [DPS22], they define Borel-Moore homology for admissible indgeometric stacks.

   • Theorem C: They construct a lax symmetric monoidal functor $H^{\text{BM}}_*(-; \mathbb{Q})$ from the category of correspondences of admissible indgeometric stacks to pro-objects in modules. This theory accommodates the non-quasi-compactness by treating homology groups as topological vector spaces (limits over quasi-compact open exhaustions) and extends functoriality to locally proper morphisms via a renormalization procedure.

3. 2-Segal Structures and Hall Multiplication:
   To define the Hall multiplication, the authors utilize the Waldhausen construction $S_\bullet \text{Coh}(\widehat{X}_Z)$, which forms a 2-Segal derived stack encoding the algebra structure in correspondences.

   • Theorem B: They prove that the relevant squares involving extension stacks are pullbacks. This ensures that the correspondence maps defining the Hall product are well-behaved (representable by finitely connected, quasi-compact, lci Artin derived stacks).
   • Crucially, they show that the Hall algebra structure depends only on the formal completion $\widehat{X}_Z$ as a formal scheme, not on the ambient surface $X$.

Key Contributions and Results

• Construction of Nilpotent COHA: The primary result is Theorem D, which establishes the existence of a unital associative topological algebra structure on the Borel-Moore homology $H^{\text{BM}}_*(\text{Coh}^{\text{nil}}_{\text{ps}}(\widehat{X}_Z); \mathbb{Q})$. The multiplication is defined via the correspondence:
  $$\text{Coh}^{\text{nil}}_{\text{ps}}(\widehat{X}_Z) \times \text{Coh}^{\text{nil}}_{\text{ps}}(\widehat{X}_Z) \xleftarrow{\partial_0 \times \partial_2} S_2 \text{Coh}^{\text{nil}}_{\text{ps}}(\widehat{X}_Z) \xrightarrow{\partial_1} \text{Coh}^{\text{nil}}_{\text{ps}}(\widehat{X}_Z)$$
  where the product is $p_* q^!$.

• Functoriality and Independence:

  • The algebra is functorial with respect to closed immersions $Z' \subset Z$ and transformations of the motivic formalism.
  • It is invariant under isomorphisms of formal completions: an isomorphism $\widehat{X}_Z \cong \widehat{X'}_{Z'}$ induces an isomorphism of topological algebras. This allows global computations to be reduced to local ones depending only on the formal neighborhood.

• 0-Dimensional Case and W-Algebras:
  In Section 6, under the assumption that the pushforward $H^{\text{BM}}_*(Z) \to H^{\text{BM}}_*(X)$ is injective (satisfied for minimal resolutions of ADE singularities and certain elliptic surfaces), the authors provide an explicit characterization of the 0-dimensional nilpotent COHA.

  • Theorem 6.7: They prove that the 0-dimensional nilpotent COHA is isomorphic to a specific subalgebra of the W-algebra associated with the pair $(X, Z)$, specifically $H_{X,Z;0} \cong W_+(\widehat{X}_Z)$. This connects the geometric construction to the algebraic structures studied in [MMSV23].

Significance and Claims
The paper claims to provide the foundational framework for nilpotent cohomological Hall algebras, generalizing the construction from quivers to surfaces. Its significance lies in:

1. Generalization: It extends the theory of COHAs to the setting of sheaves supported on arbitrary proper curves (possibly singular) on smooth surfaces, generalizing the global nilpotent cone.
2. Intrinsic Construction: It offers an intrinsic construction of the algebra of cohomological Hecke operators, independent of the ambient surface's global geometry, relying solely on the formal neighborhood of the curve.
3. Bridge to Quantum Groups: By establishing the link to W-algebras in the 0-dimensional case and the relationship to preprojective COHAs (via [DPS$^+$26]), the work provides a crucial tool for understanding the precise relationship between surface COHAs and quantum groups/Yangians.
4. Technical Advancement: It advances the theory of motivic Borel-Moore homology to the class of admissible indgeometric stacks, enabling the handling of non-quasi-compact moduli problems in a rigorous derived setting.

The authors note that the results are intended to be used in the companion paper [DPS$^+$26] to answer specific questions regarding the relationship between COHAs of Kleinian singularities and preprojective algebras, but they do not claim to solve the full presentation problem for arbitrary surfaces within this text, noting that the 0-dimensional case is the most explicitly understood.