Cleft Extensions of Rings and Their Impact on Singularity Categories and Gorenstein Homological Properties

Kernkonzepte

This paper investigates the homological properties of cleft extensions of rings, demonstrating their impact on Gorensteinness, singularity categories, and big singularity categories, ultimately leading to equivalences under specific conditions.

Zusammenfassung

• Bibliographic Information: Kostas, P. (2024). Cleft extensions of rings and singularity categories. arXiv preprint arXiv:2409.07919v2.
• Research Objective: This paper aims to study the behavior of singularity categories and Gorenstein homological properties under cleft extensions of rings, extending previous research focused on ring homomorphisms.
• Methodology: The paper utilizes the framework of cleft (co)extensions of abelian categories, focusing on the homological properties of perfect and nilpotent endofunctors associated with these extensions.
• Key Findings: The study reveals that under a cleft extension of rings, the Gorensteinness of one ring implies the Gorensteinness of the other. Furthermore, specific conditions related to the vanishing of certain derived functors lead to equivalences between the singularity categories and big singularity categories of the rings involved in the cleft extension.
• Main Conclusions: The research provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings, unifying a range of existing results and offering new insights into the relationship between the homological properties of rings connected through cleft extensions.
• Significance: This work contributes significantly to the field of homological algebra by providing a deeper understanding of the interplay between cleft extensions of rings and their impact on singularity categories and Gorenstein properties.
• Limitations and Future Research: The paper primarily focuses on cleft extensions where a specific endofunctor is perfect and nilpotent. Future research could explore the implications of relaxing these conditions or investigating other categorical constructions beyond cleft extensions.

Statistiken

gl. dimΛ < ∞ means the global dimension of ring Λ is finite.
Dsg(Λ) represents the singularity category of a Noetherian ring Λ.
Kac(Inj-Λ) denotes the big singularity category of ring Λ.
Gproj-Λ represents the stable category of Gorenstein projective modules over Λ.
F represents the endofunctor − ⊗Γ M.
F' represents the endofunctor HomΓ(M, −).

Zitate

"Singularity categories are nowadays heavily studied in algebra and geometry."
"The singularity category of a Noetherian ring Λ, denoted by Dsg(Λ), [...] is a measure of regularity."
"A fundamental theorem of Buchweitz states that there is a fully faithful functor i: Gproj-Λ →Dsg(Λ) where the left-hand side denotes the stable category of the Gorenstein projective modules over Λ."
"The aim of this paper is to study the behaviour of the above triangulated categories under a cleft extension of rings."
"This is in the spirit of Oppermann-Psaroudakis-Stai [45], where the same problem was investigated under a ring homomorphism."

Wichtige Erkenntnisse aus

Cleft extensions of rings and singularity categories

by Panagiotis K... um arxiv.org 11-01-2024

https://arxiv.org/pdf/2409.07919.pdf

Tiefere Fragen

How do the findings of this paper extend to other algebraic structures beyond rings, such as differential graded algebras or categories?

This paper focuses on cleft extensions of rings and their singularity categories, utilizing the framework of abelian categories. While the specific results are stated for rings, the underlying principles and techniques hint at potential extensions to more general algebraic structures:

Differential Graded Algebras (DGAs): DGAs are a natural generalization of rings, incorporating chain complexes and differentials. Many concepts from ring theory, including singularity categories, have analogues in the DGA world. It's plausible that under suitable adaptations, the notion of a cleft extension could be defined for DGAs. The key would be to establish analogous homological properties for these extensions and investigate if similar equivalences between singularity categories hold.

Categories: The paper already utilizes the language of abelian categories, which provides a level of abstraction beyond rings. This suggests that the results could be further generalized. One possible direction is to consider cleft extensions within more general categorical frameworks, such as exact categories or triangulated categories. The challenge would lie in finding appropriate generalizations of Gorensteinness, perfect endofunctors, and singularity categories within these settings.
Key Challenges and Considerations:

Homological Algebra: Extending the results would require developing the necessary homological algebra machinery (e.g., derived categories, Gorenstein projective objects) for the specific algebraic structure in question.
Perfectness Conditions: The notion of a perfect endofunctor plays a crucial role. Finding suitable analogues of this condition in other settings would be essential.
Geometric Interpretation: If the goal is to connect with algebraic geometry, a clear geometric interpretation of cleft extensions and their singularity categories in the new context would be highly desirable.

Could there be scenarios where a cleft extension of rings does not lead to an equivalence of singularity categories, even when the endofunctor F is not perfect and nilpotent?

Yes, absolutely. The conditions of F being perfect and nilpotent are sufficient to establish an equivalence of singularity categories in the context of cleft extensions of rings, as stated in Theorem B(ii). However, these conditions are not necessary.
Here's why we might not get an equivalence even without those conditions:

Singularity Categories are Complex: Singularity categories capture subtle information about the resolutions of objects in an abelian category. Even small changes in the structure of the rings involved in a cleft extension can lead to significant differences in their singularity categories.
Endofunctor's Influence: While not perfect and nilpotent, the endofunctor F could still have a nontrivial impact on the homological properties of the categories involved. This influence might prevent a straightforward equivalence.
Possible Scenarios:

F Affects Resolutions: Even if F doesn't satisfy the perfectness conditions, it might still alter the structure of projective or injective resolutions in a way that leads to different singularity categories.
Hidden Structure: There might be additional structural relationships between the rings Γ and Λ in the cleft extension that are not fully captured by the endofunctor F alone. These hidden relationships could influence the singularity categories.
In essence, the relationship between cleft extensions and singularity categories is intricate. The conditions presented in the paper provide a powerful tool for establishing equivalences, but they don't encompass all possible scenarios.

What are the implications of these findings for understanding the geometric significance of singularity categories in algebraic geometry, particularly in the context of derived categories of coherent sheaves?

The findings of this paper, while focused on algebraic notions, have intriguing implications for understanding the geometric significance of singularity categories in algebraic geometry:

New Tools for Equivalences: The paper provides new tools for establishing equivalences between singularity categories, particularly in the context of cleft extensions. This is significant because many geometric constructions can be viewed through the lens of algebraic extensions.
Relating Singularities: In algebraic geometry, singularity categories are a powerful way to study singularities of algebraic varieties. The paper's results suggest that cleft extensions could provide a framework for relating the singularities of different varieties. For example, if two varieties are related by a suitable geometric construction that corresponds to a cleft extension algebraically, their singularity categories might be equivalent, reflecting a deeper geometric connection.
Derived Categories of Coherent Sheaves: The derived category of coherent sheaves on a variety is a fundamental object of study in algebraic geometry, and its structure is intimately connected to the singularity category. The paper's focus on Gorenstein properties and perfect endofunctors could provide insights into the structure of derived categories in cases where geometric constructions induce cleft extensions at the level of sheaves.
Potential Geometric Applications:

Blow-ups and Resolutions: Blow-ups are fundamental operations in algebraic geometry used to resolve singularities. It would be interesting to explore if certain blow-up constructions can be understood as cleft extensions algebraically, and if so, what the implications are for the singularity categories of the involved varieties.
Fiber Products and Quotients: Fiber products and quotients are common ways to construct new varieties from old ones. Investigating if these constructions can be related to cleft extensions could provide insights into how singularities behave under these operations.
Overall, the paper's findings suggest a promising avenue for utilizing algebraic techniques related to cleft extensions to further our understanding of the geometric meaning of singularity categories and their connection to the geometry of algebraic varieties.