Roos Axiom (AB4*-n) Holds for the Category of Quasi-coherent Sheaves on a Quasi-compact Semi-separated Scheme or a Noetherian Scheme of Finite Krull Dimension

Core Concepts

The Roos axiom, a condition concerning the vanishing of higher derived functors of infinite products in abelian categories, holds for the category of quasi-coherent sheaves on specific types of schemes, namely quasi-compact semi-separated schemes and Noetherian schemes of finite Krull dimension.

Abstract

This research paper investigates the validity of the Roos axiom (AB4*-n) within the context of quasi-coherent sheaves on specific types of schemes. The Roos axiom, a weaker form of Grothendieck's axiom AB4*, posits the finite homological dimension of derived functors of infinite products in specific abelian categories.

The paper focuses on two types of schemes: quasi-compact semi-separated schemes and Noetherian schemes of finite Krull dimension. The authors employ two distinct approaches to demonstrate the validity of the Roos axiom in these scenarios.

The first approach utilizes the Čech coresolution technique. This method involves constructing a specific resolution of a quasi-coherent sheaf and analyzing the derived functors of the direct image functor applied to this resolution. By leveraging the properties of the Čech coresolution and the direct image functor, the authors establish the vanishing of higher derived functors of infinite products, thereby proving the Roos axiom.

The second approach revolves around the concept of very flat quasi-coherent sheaves. The authors demonstrate the existence of a generator of finite projective dimension within the category of quasi-coherent sheaves on a quasi-compact semi-separated scheme. This finding, coupled with the properties of very flat sheaves, provides an alternative proof of the Roos axiom.

The paper highlights the significance of the Roos axiom in understanding the homological properties of categories of quasi-coherent sheaves. The results presented contribute to the broader study of sheaf theory and its applications within algebraic geometry.

• Bibliographic Information: Positselski, L. (2024). Roos Axiom Holds for Quasi-coherent Sheaves. arXiv:2407.13651v2 [math.AG].
• Research Objective: To prove that the Roos axiom (AB4*-n) holds for the category of quasi-coherent sheaves on quasi-compact semi-separated schemes and Noetherian schemes of finite Krull dimension.
• Methodology: The paper employs two main approaches: (1) Analysis of the Čech coresolution of quasi-coherent sheaves and the derived functors of the direct image functor. (2) Demonstration of the existence of a generator of finite projective dimension in the category of quasi-coherent sheaves on a quasi-compact semi-separated scheme, utilizing the concept of very flat quasi-coherent sheaves.
• Key Findings: The paper successfully proves that the Roos axiom holds for the category of quasi-coherent sheaves on both quasi-compact semi-separated schemes and Noetherian schemes of finite Krull dimension.
• Main Conclusions: The established validity of the Roos axiom in these specific categories of sheaves contributes significantly to the understanding of their homological properties.
• Significance: The research enhances the theoretical framework of sheaf theory and its applications within algebraic geometry, particularly in studying schemes with specific properties.
• Limitations and Future Research: The paper focuses on specific types of schemes. Further research could explore the validity of the Roos axiom in broader contexts or for other categories of sheaves.

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Roos axiom holds for quasi-coherent sheaves

by Leonid Posit... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2407.13651.pdf

Deeper Inquiries

What are the implications of the Roos axiom holding for these specific categories of sheaves in other areas of mathematics, such as representation theory or homological algebra?

Answer:
The Roos axiom, a condition concerning the vanishing of higher derived functors of infinite direct products, has significant implications when it holds for categories of quasi-coherent sheaves, particularly in representation theory and homological algebra:
Representation Theory:

Representation categories of quivers: The category of quasi-coherent sheaves on an affine scheme is equivalent to the category of modules over its coordinate ring. This connects the Roos axiom to representation theory of quivers, as modules over path algebras of quivers are central objects of study. The axiom's presence can simplify homological computations and provide insights into the structure of these representation categories.
Derived equivalences and tilting theory: The Roos axiom can be preserved under derived equivalences, which are fundamental in tilting theory. This theory aims to relate different categories with 'tilting objects,' and the axiom's validity can help establish such equivalences and transfer homological information between seemingly disparate representation-theoretic contexts.
Homological Algebra:

Finiteness conditions on derived categories: The Roos axiom, particularly the AB4*-n condition, implies a certain "finiteness" property of the derived category of quasi-coherent sheaves. This has implications for understanding the structure of these derived categories, especially when studying t-structures and their hearts.
Cohomology of sheaves and vanishing theorems: The axiom can be used to deduce vanishing theorems for higher cohomology groups of sheaves, which are crucial in algebraic geometry and topology. These vanishing results have implications for understanding geometric properties of schemes and varieties.
Limits and colimits: The Roos axiom's validity simplifies the study of limits and colimits in categories of quasi-coherent sheaves. This is particularly relevant in understanding derived functors involving limits, such as the derived functor of inverse limit, which has applications in studying coherent cohomology.
Overall, the Roos axiom's presence in the context of quasi-coherent sheaves provides a powerful tool for homological computations and structural insights, bridging algebraic geometry, representation theory, and homological algebra.

Could there be alternative methods, beyond the Čech coresolution and very flat sheaves, to prove or disprove the Roos axiom for quasi-coherent sheaves on other types of schemes?

Answer:
Yes, beyond Čech coresolutions and very flat sheaves, several alternative methods could be employed to investigate the Roos axiom for quasi-coherent sheaves on various schemes:
1. Spectral Sequences:

Hypercohomology spectral sequences: These sequences relate the derived functors of a composition of functors to the derived functors of the individual functors. They could be used to analyze the derived functors of infinite direct products by decomposing them into more manageable pieces.
Leray spectral sequences: For a morphism of schemes, these sequences connect sheaf cohomology on the source and target. They could be helpful when studying the Roos axiom for schemes with suitable morphisms to schemes where the axiom is already known to hold.
2. Geometric Techniques:

Ample families of line bundles: For schemes with ample families of line bundles, one could leverage the properties of these bundles to study the behavior of infinite direct products of quasi-coherent sheaves. This approach could be particularly fruitful for projective schemes.
Koszul complexes: These complexes provide resolutions for sheaves on schemes, and their structure could be exploited to analyze the derived functors of infinite direct products.
3. Model-Categorical Methods:

Model structures on categories of sheaves: By equipping categories of quasi-coherent sheaves with appropriate model structures, one could study the Roos axiom using homotopy-theoretic tools. This approach could provide a more abstract and flexible framework for analyzing the axiom.
4. Co-Contra Correspondence (Beyond the "Naive" Version):

Coderived and contraderived categories: As mentioned in the context, exploring the relationship between coderived categories of exact categories with exact coproducts and contraderived categories of exact categories with exact products could offer deeper insights. This could lead to more refined versions of the co-contra correspondence, potentially applicable to a broader class of schemes.
Disproving the Roos Axiom:

Counterexamples: To disprove the Roos axiom for specific types of schemes, one could try to construct explicit counterexamples. This would involve finding families of quasi-coherent sheaves for which the higher derived functors of their infinite direct product do not vanish.
The choice of method would depend on the specific type of scheme and the available tools. It's worth noting that the Roos axiom might not hold for all types of schemes, and finding counterexamples or characterizing the schemes for which it holds remains an active area of research.

How does the concept of "very flatness" in sheaf theory relate to analogous notions of "flatness" in other mathematical structures, and what insights can be drawn from these connections?

Answer:
The concept of "very flatness" in sheaf theory, as introduced in the context of quasi-coherent sheaves, exhibits strong connections to analogous notions of "flatness" found in other mathematical structures. These connections provide valuable insights into the nature of flatness and its role in homological algebra and beyond.
1. Module Theory:

Flat modules: The foundational notion of a flat module over a ring captures the property that tensoring with the module preserves exact sequences. Very flat modules, as defined in the context, are a special class of flat modules characterized by their vanishing Ext groups with contraadjusted modules. This highlights a finer distinction within the realm of flatness, emphasizing homological properties beyond the preservation of exactness.
2. Commutative Algebra:

Flat morphisms of rings: A ring homomorphism is flat if it makes the target ring a flat module over the source ring. Similarly, a morphism of schemes is very flat if it locally corresponds to very flat ring homomorphisms. This parallel underscores the geometric manifestation of very flatness, reflecting the local nature of schemes.
3. Homotopy Theory:

Projective objects: In abstract homotopy theory, projective objects play a crucial role in resolutions and homological constructions. Very flat sheaves, while not necessarily projective themselves, share a key property with projective objects: they are direct summands of filtered objects with "simple" quotients (analogous to free modules in module theory). This connection suggests that very flatness can be viewed as a weaker form of projectivity, still potent enough for certain homological applications.
Insights from the Connections:

Hierarchical structure of flatness: The concept of very flatness reveals a hierarchical structure within the notion of flatness, with very flat objects forming a distinguished subclass of flat objects. This hierarchy reflects the interplay between different levels of homological vanishing conditions.
Geometric interpretation of homological properties: The connection between very flat modules and very flat morphisms of schemes provides a geometric interpretation of homological properties. It highlights how abstract algebraic conditions translate into concrete geometric properties of maps between spaces.
Tools for homological algebra: The notion of very flatness, like its counterparts in other areas, provides valuable tools for homological algebra. The existence of very flat resolutions simplifies computations and facilitates the study of derived functors.
In summary, the concept of "very flatness" in sheaf theory enriches the understanding of flatness by revealing its nuanced nature and its connections across different mathematical domains. It underscores the importance of flatness as a unifying theme in algebra, geometry, and homotopy theory.