A note on quasi-perfect morphisms

This paper establishes a new characterization of regular Noetherian algebraic spaces via quasi-perfect blowups and demonstrates that the quasi-perfectness of proper morphisms is an étale-local property, thereby proving that the locus of such points is Zariski open.

The Gist

Imagine you are an architect trying to understand the structural integrity of a massive, complex building (which mathematicians call an "algebraic space"). Sometimes, the building looks fine from the outside, but up close, the walls are crumbling, or the foundation is shaky. In mathematics, we call a building "regular" if it is perfectly smooth and well-constructed everywhere. If it has cracks or weird bumps, it's "singular" or "irregular."

This paper is like a new set of tools for architects to check if a building is truly "regular" and to understand how the quality of the building changes when you move between different parts of it.

Here is a breakdown of the paper's two main discoveries using simple analogies:

1. The "Demolition and Rebuild" Test (Characterizing Regularity)

The Problem: How do you know if a specific point in your building is perfectly smooth (regular)? Usually, you have to look at the microscopic details of the materials, which is hard.

The New Tool: The authors propose a clever test involving a blowup. In construction terms, imagine you find a specific spot (a "closed point") on the building. You decide to demolish that exact spot and rebuild it, creating a new, slightly larger structure around it (this is the "blowup").

• The Analogy: Think of the building as a piece of fabric. If you have a tiny snag (a singular point) and you try to iron it out by stretching the fabric around it (the blowup), the fabric might tear or behave weirdly. But if the fabric was already smooth, the ironing process goes perfectly.
• The Discovery: The authors found that if you perform this "demolish and rebuild" on a point, and the resulting new structure behaves perfectly (mathematically, it is "quasi-perfect"), then the original point was always smooth to begin with.
• The Takeaway: You don't need a microscope to check if a point is smooth. Just try to "blow it up." If the result is perfect, the point was perfect. If the result is messy, the point was flawed. This gives a brand new way to define what a "perfect" building looks like.

2. The "Local Inspection" Rule (Detecting Quality Locally)

The Problem: Suppose you have a bridge connecting two cities (a "proper morphism"). You want to know if the bridge is structurally sound (quasi-perfect) for the whole journey. Do you have to inspect the entire bridge at once? That's impossible.

The New Tool: The authors show that you can check the bridge's quality by looking at tiny, local samples taken from different spots along the route.

• The Analogy: Imagine the bridge is a long, winding road. Instead of driving the whole way to see if it's safe, you can take a sample of the asphalt from a specific mile marker, or look at the road through a microscope at a specific spot, or even look at a "frozen" version of that spot (a completion).
• The Discovery: If the bridge is safe at every single one of these tiny, local checkpoints (specifically at the "local rings," "completions," or "Henselizations"), then the entire bridge is safe. Conversely, if the bridge fails at even one of these local spots, the whole thing is flawed.
• The "Open" Secret: Even cooler, they found that the spots where the bridge is safe form a continuous, unbroken area. It's not a scattered collection of safe spots; if you find a safe spot, the spots immediately around it are also safe. This means the "safe zone" is always a nice, open area on the map.

Why Does This Matter?

In the world of algebraic geometry, "regularity" is the gold standard. It means things behave nicely and predictably. "Singularities" (irregularities) are where things get messy and hard to solve.

• For Mathematicians: This paper provides a new way to prove that a space is regular without doing heavy lifting. It also solves a long-standing mystery: "Can we tell if a complex structure is good just by looking at its tiny, local pieces?" The answer is yes, provided the structure connects the pieces properly (is "proper").
• The Big Picture: Just as a master builder can tell if a house is sound by checking the foundation and the local joints, these mathematicians have proven that you can determine the health of a complex mathematical universe by inspecting its smallest, local neighborhoods.

In summary:

1. Test for Smoothness: To see if a point is perfect, try to "blow it up." If the result is perfect, the point was perfect.
2. Check the Whole by the Parts: To see if a connection is perfect, just check the tiny, local rings. If they are all good, the whole connection is good, and the "good" spots will always form a nice, open neighborhood.

This work helps mathematicians navigate the messy, complex landscapes of algebraic spaces with a clearer, more reliable map.

Technical Summary

Here is a detailed technical summary of the paper "A Note on Quasi-Perfect Morphisms" by Timothy De Deyn, Pat Lank, and Kabeer Manali-Rahul.

1. Problem Statement and Context

The paper addresses the behavior of quasi-perfect morphisms between Noetherian algebraic spaces.

• Background: For a quasi-compact quasi-separated morphism $f: Y \to X$, the derived pushforward $Rf_*$ admits a right adjoint $f^\times$ (by Neeman's Brown Representability Theorem). However, $f^\times$ does not always preserve small coproducts.
• Definition: A morphism $f$ is quasi-perfect if $f^\times$ preserves small coproducts on the derived category of quasi-coherent sheaves ($D_{qc}$). Equivalently, $Rf_*$ maps perfect complexes to perfect complexes ($Rf_* \text{Perf}(Y) \subseteq \text{Perf}(X)$).
• The Gap: While quasi-perfectness is well-understood globally (e.g., proper morphisms of finite Tor dimension), its local behavior is less clear. Specifically, it was unknown how the quasi-perfectness of a morphism at local rings (completions, Henselizations) relates to the global property, and whether the locus of points where a morphism is quasi-perfect is open. Additionally, a characterization of regularity using blowups at closed points was missing in the literature for algebraic spaces.

2. Methodology

The authors employ techniques from derived algebraic geometry and triangulated category theory, specifically focusing on:

• Compact Generation: Utilizing the fact that $D_{qc}(X)$ is compactly generated by perfect complexes.
• Flat Descent: Using fpqc, fppf, and étale covers to check properties locally.
• Local-to-Global Techniques: Reducing global questions about proper morphisms to questions over local rings (Henselizations, completions).
• Blowup Geometry: Analyzing the derived pushforward of the exceptional divisor in blowups to detect regularity.

Key technical tools include:

• Lemma 4.1: Characterizing quasi-perfectness via compact generators.
• Lemma 4.2 & 4.3: Establishing that quasi-perfectness can be checked via flat surjective covers (specifically quasi-affine morphisms).
• Stalk Analysis: Examining the perfectness of skyscraper sheaves at closed points.

3. Key Contributions and Results

A. Characterization of Regularity via Blowups

The authors provide a novel characterization of regular Noetherian algebraic spaces.

• Proposition 3.3: A Noetherian algebraic space $X$ is regular if and only if the blowup of $X$ along any closed point is a quasi-perfect morphism.
• Mechanism:
  1. They prove (Lemma 3.1) that a closed point $x$ is regular if and only if the skyscraper sheaf $i_{x,*}\mathcal{O}_{\kappa(x)}$ is a perfect complex.
  2. They show (Lemma 3.2) that if the blowup at a closed point $p$ is quasi-perfect, the exceptional divisor's pushforward forces the skyscraper sheaf at $p$ to be perfect, implying $p$ is regular.
• Significance: This is the first characterization of regularity for algebraic spaces (and even schemes) that relies solely on the quasi-perfectness of blowups at closed points, independent of the existence of a specific regular alteration.

B. Local Detection of Quasi-Perfectness

The paper establishes that for proper morphisms, quasi-perfectness is a local property detectable at specific local rings.

• Theorem 4.8: Let $f: Y \to X$ be a proper morphism of Noetherian algebraic spaces. The following are equivalent:
  1. $f$ is quasi-perfect.
  2. The base change of $f$ to $\text{Spec}(A_p)$ is quasi-perfect for all $p \in |X|$, where $A_p$ is either the Henselization ($\mathcal{O}^h_{X,p}$) or the Strict Henselization ($\mathcal{O}^{sh}_{X,p}$) of the local ring at $p$.
• Method: The proof reduces the global question to local rings using étale covers and the fact that Henselization/Completion are faithfully flat. It leverages the fact that for proper morphisms, the pushforward of a compact generator remains bounded and coherent, allowing local checks to imply global results.

C. The Quasi-Perfect Locus is Open

A major consequence of the local detection theorem is the topological nature of the locus where a morphism is quasi-perfect.

• Corollary 4.9: For a proper morphism $f: Y \to X$, the set of points $p \in X$ where the base change to the local ring at $p$ is quasi-perfect forms a Zariski open subset of $X$.
• Significance: This result is new even for schemes. It implies that the "quasi-perfect locus" is well-behaved topologically, which is crucial for studying singularities and moduli problems.

D. Descent of Regularity Properties

The authors apply these results to descent theory.

• Proposition 4.11: If $f: Y \to X$ is a proper surjective morphism of Noetherian algebraic spaces, and $Y$ has a non-empty open regular locus (i.e., $Y$ is $J$-0), then $X$ also has a non-empty open regular locus, provided $X$ has a codimension 0 point with a reduced étale local ring.
• Mechanism: This uses the "quasi-perfect locus" $S$ from Corollary 4.9. Since $f$ is quasi-perfect over $S$, and $Y$ is regular over a dense open set, the authors show $X$ must be regular over a dense open subset of $S$.

4. Significance and Impact

• New Characterization of Regularity: The link between blowups at closed points and regularity offers a new tool for verifying regularity in algebraic spaces without constructing global resolutions.
• Local-to-Global Principle: Theorem 4.8 bridges the gap between local ring theory and global derived categories for proper morphisms, showing that complex global properties (quasi-perfectness) are determined by local data (Henselizations).
• Topological Control: Proving the quasi-perfect locus is Zariski open provides a foundational result for future work on the Minimal Model Program (MMP) for algebraic spaces, particularly in characteristic zero where MMP has recently been established.
• Independence from Global Alterations: Unlike previous results that required the existence of a regular alteration to deduce regularity, this work provides intrinsic characterizations.

5. Limitations and Scope

• Properness Assumption: The local detection results (Theorem 4.8) and the openness of the locus (Corollary 4.9) rely heavily on the morphism being proper. This is necessary to ensure the pushforward of perfect complexes remains bounded and coherent. The authors note that for separated morphisms of finite type between Noetherian schemes, quasi-perfectness often implies properness, making the restriction mild in many contexts.
• Noetherian Setting: The results are framed within the context of Noetherian algebraic spaces.

In summary, this note refines the understanding of quasi-perfect morphisms by establishing their local detectability for proper maps, characterizing regularity via blowups, and proving the openness of the quasi-perfect locus, thereby providing essential tools for the study of singularities in algebraic spaces.