Cotorsion Pairs and Tor-Pairs over Commutative Noetherian Rings: A Classification Theorem and Its Applications

Core Concepts

This research paper classifies specific types of cotorsion pairs and Tor-pairs over commutative noetherian rings, establishing a connection between them through duality and exploring their applications in characterizing modules and rings.

Abstract

Herbera, D., Hrbek, M., & Le Gros, G. (2024). Cotorsion pairs and Tor-pairs over commutative noetherian rings. arXiv:2411.04514v1 [math.AC].

This paper aims to classify cotorsion pairs cogenerated by pure-injective modules of finite injective dimension and Tor-pairs generated by modules of finite flat dimension over commutative noetherian rings. The authors investigate the relationship between these two structures and their implications for module and ring characterization.

Key Insights Distilled From

Cotorsion pairs and Tor-pairs over commutative noetherian rings

by Dolors Herbe... at arxiv.org 11-08-2024

https://arxiv.org/pdf/2411.04514.pdf

Deeper Inquiries

How can the classification of cotorsion pairs and Tor-pairs be extended to non-commutative noetherian rings, and what challenges arise in this more general setting?

Extending the classification of cotorsion pairs and Tor-pairs to non-commutative Noetherian rings presents significant challenges. Here's a breakdown of potential approaches and the hurdles involved:
Challenges in the Non-Commutative Setting:

Loss of Prime Spectrum: The prime spectrum, a fundamental tool in commutative algebra, doesn't have a direct analogue in the non-commutative world. The lack of a well-behaved notion of "prime ideals" makes it difficult to define local depth and utilize tools like localization, which are crucial in the commutative classification.

Complexity of Injective Modules: Injective modules over non-commutative Noetherian rings can be far more intricate. The structure of injective modules is intimately tied to the classification of cotorsion pairs, as pure-injective modules play a central role.

Asymmetry of Torsion Pairs: Unlike the commutative case, Tor-pairs over non-commutative rings are not symmetric. This asymmetry adds complexity to their classification.

Possible Approaches and Their Limitations:

Adapting Depth and Grade: One approach could involve finding suitable generalizations of depth and grade for non-commutative rings. However, defining these notions in a way that captures the necessary homological information and behaves well with respect to localization (or its analogues) is a major obstacle.

Alternative Homological Invariants: Exploring alternative homological invariants that capture the relevant properties of modules over non-commutative Noetherian rings could be fruitful. This might involve developing new dimension theories or studying different Ext and Tor functors.

Categorical Methods: Employing more abstract categorical methods, such as derived categories and triangulated categories, might offer a way to circumvent some of the difficulties. However, translating results back to the module-theoretic level can be challenging.

Specific Considerations:

Artin Algebras: For certain classes of non-commutative Noetherian rings, such as Artin algebras, more specialized techniques might be applicable. Representation theory of quivers could provide valuable tools in this context.

Semiprime Rings: Focusing on semiprime Noetherian rings, where the intersection of all prime ideals is zero, might offer a more tractable setting.

Could there be alternative characterizations of the studied cotorsion and Tor-pairs, perhaps using different homological invariants or categorical constructions?

Yes, exploring alternative characterizations of cotorsion and Tor-pairs using different homological invariants or categorical constructions is a promising avenue for research. Here are some potential directions:
1. Dimension Theories Based on Approximations:

Instead of focusing solely on projective, injective, or flat dimensions, one could investigate dimensions defined in terms of approximations by modules from specific classes. For example, Gorenstein projective and Gorenstein injective dimensions have proven useful in characterizing certain cotorsion pairs.
2. Relative Homological Algebra:

Employing techniques from relative homological algebra, where resolutions are taken with respect to a fixed class of modules, could lead to new insights. This approach allows for a more refined analysis of homological properties.
3. Model-Theoretic Approaches:

Model theory, particularly the study of definable sets and imaginaries, has found applications in module theory. It might be possible to characterize certain cotorsion and Tor-pairs in terms of model-theoretic invariants.
4. Derived and Triangulated Categories:

Cotorsion pairs and Tor-pairs have natural interpretations in the derived category. Studying their properties within this framework, using tools like t-structures and Bousfield localizations, could reveal new characterizations.
5. Homological Dimensions of Complexes:

Instead of restricting to modules, one could consider homological dimensions of complexes. This broader perspective might lead to a more comprehensive understanding of cotorsion and Tor-pairs.
Example: Gorenstein Homological Algebra:

Gorenstein projective, injective, and flat modules have associated dimensions that have been successfully used to characterize certain cotorsion pairs. These dimensions capture finiteness properties related to resolutions by these "Gorenstein" modules.

What are the implications of the established duality between cotorsion and Tor-pairs for other areas of mathematics, such as algebraic geometry or representation theory of quivers?

The duality between cotorsion and Tor-pairs, particularly in the context of commutative Noetherian rings, has intriguing implications for other areas of mathematics:
Algebraic Geometry:

Sheaf Cohomology and Derived Categories: The duality could potentially shed light on the relationship between sheaf cohomology and the derived category of coherent sheaves on a scheme. Cotorsion pairs and Tor-pairs have natural interpretations in the derived category, and their duality might provide insights into the interplay between different cohomology theories.

Singularity Theory: Depth and other homological invariants are closely related to the singularities of algebraic varieties. The classification of cotorsion and Tor-pairs using these invariants could lead to a deeper understanding of singularity types and their properties.

Representation Theory of Quivers:

Classifying Representations: Cotorsion pairs have proven valuable in classifying representations of quivers and finite-dimensional algebras. The duality with Tor-pairs might provide new tools for understanding the structure of module categories over these algebras.

Tilting Theory: Tilting theory, which plays a central role in representation theory, is closely related to cotorsion pairs. The duality with Tor-pairs could potentially lead to new tilting-like constructions and results.

Other Areas:

Commutative Algebra: The duality provides a new perspective on classical concepts in commutative algebra, such as depth, grade, and Koszul complexes. It could inspire further research into the homological properties of commutative rings.

Homological Algebra: The interplay between cotorsion and Tor-pairs highlights the rich structure of module categories and their homological invariants. It could motivate the development of new techniques and results in homological algebra.

Key Points:

The duality provides a bridge between different areas of mathematics, allowing for the transfer of ideas and techniques.
It offers a new perspective on classical concepts and could lead to a deeper understanding of their interconnections.
The implications are still being explored, and further research is needed to fully grasp the potential of this duality.