Delooping Levels and Their Relationship to Other Homological Invariants in Artin Algebras

Основные понятия

This research paper introduces the concept of global delooping level (Dell) for Artin algebras, investigates its relationship to other homological invariants like finitistic dimension and Igusa-Todorov functions, and demonstrates its application in specific cases like Gorenstein and truncated path algebras.

Аннотация

Barrios, M., Lanzilotta, M., & Mata, G. (2024). Delooping levels. arXiv:2410.16422v1 [math.RT].

This paper aims to introduce a new homological invariant called "global delooping level" (Dell) for Artin algebras and explore its relationship with other established homological dimensions, such as finitistic dimension, delooping level (dell), and Igusa-Todorov functions.

Ключевые выводы из

Delooping levels

by Marcos Barri... в arxiv.org 10-23-2024

https://arxiv.org/pdf/2410.16422.pdf

Дополнительные вопросы

How does the concept of global delooping level relate to other homological invariants in more general algebraic structures beyond Artin algebras?

While the provided context focuses on Artin algebras, the concept of delooping level, particularly the global delooping level (Dell), can potentially be extended and studied in the context of more general algebraic structures. Here's how it might relate to other homological invariants:
1. Beyond Artin Algebras: The definition of Dell relies on the syzygy functor (Ω) and the additive structure of the module category. These concepts exist in more general settings like:
* **Noetherian Rings:** The paper already mentions Gelinas' work on Noetherian semiperfect rings. For a Noetherian ring, one could define Dell similarly and explore its relationship with invariants like **Krull dimension, injective dimension, and global dimension**.
* **Abelian Categories:** Syzygies and projective resolutions can be defined in any abelian category. Dell could be studied in this abstract setting, relating it to the **structure of the projective objects** and the **finitistic dimension of the category**.

2. Connections to Other Invariants:
* **Gorenstein Homological Algebra:** The context highlights a strong connection between Dell and Gorenstein projective modules for Gorenstein algebras. This suggests exploring Dell in categories with **relative homological algebra** frameworks, like Gorenstein projective/injective modules over Noetherian rings.
* **K-Theory:** The definition of Igusa-Todorov functions uses the Grothendieck group (K₀). Dell might have connections to **higher K-theory groups** and provide insights into the **stable module category**.
* **Derived Categories:** Delooping level, as the name suggests, might have interpretations in the **derived category** of an algebra or a more general abelian category. It could relate to the **structure of the singularity category** (the Verdier quotient of the bounded derived category by the perfect complexes).

Challenges:
* **Existence of Projective Resolutions:** In more general settings, projective resolutions might not always exist. One might need to work with **approximations** or restrict to suitable subcategories.
* **Finiteness Conditions:** The finiteness of Dell is an important consideration. Additional conditions on the algebraic structure might be needed to ensure well-behaved delooping levels.

Could there be alternative definitions or modifications to the concept of Dell that might lead to stronger or more general results regarding its relationship with finitistic dimension?

Yes, there are several potential modifications or alternative perspectives on Dell that could lead to a stronger connection with finitistic dimension:
1. Relative Delooping Level:
* **Idea:** Instead of considering all modules, define a relative delooping level with respect to a subcategory. For example, dell_C(M) could be the minimal n such that Ωⁿ(M) is a direct summand of Ωⁿ⁺¹(N) for some N in the subcategory C.
* **Potential:** Choosing suitable subcategories (e.g., modules of finite projective dimension) might yield a tighter relationship with finitistic dimension.

2. Asymptotic Delooping Level:
* **Idea:** Instead of requiring Ωⁿ(M) to be a direct summand, consider the condition that Ωⁿ(M) is a direct summand of Ω^m(N) for some m >> n (m much larger than n).
* **Potential:** This relaxed condition might be easier to satisfy and still provide information about the eventual behavior of syzygies.

3. Delooping Level of Complexes:
* **Idea:** Define delooping level for objects in the derived category, not just modules. This would allow for a more flexible notion of "delooping."
* **Potential:** This could lead to connections with the **Rouquier dimension** of the derived category, which is known to bound the finitistic dimension in some cases.

4. Delooping Level and Dimensions of Triangulated Categories:
* **Idea:** Explore connections between Dell and dimensions of triangulated categories, such as **generation time** or **Orlov spectrum**.
* **Potential:** These dimensions capture the complexity of the triangulated structure and might provide bounds or characterizations of Dell.

5. Weaker Conditions than Direct Summands:
* **Idea:** Instead of requiring Ωⁿ(M) to be a direct summand, consider weaker conditions like being a **submodule** or a **factor module** of some Ωⁿ⁺¹(N).
* **Potential:** This could lead to a more refined hierarchy of delooping levels and potentially stronger relationships with finitistic dimension.

What are the potential implications of understanding Dell in the context of classifying Artin algebras or studying their representation-theoretic properties?

Understanding the global delooping level (Dell) could have significant implications for the classification of Artin algebras and the study of their representations:
1. New Classes of Algebras:
* **Dell as an Invariant:** Dell provides a new numerical invariant for Artin algebras. Classifying algebras with specific Dell values (e.g., Dell = 0, Dell = 1, finite Dell) could lead to interesting families of algebras with shared representation-theoretic properties.
* **Connections to Existing Classes:** The context shows that Dell is related to Gorenstein algebras, truncated path algebras, and algebras of finite finitistic dimension. A deeper understanding of Dell might reveal further connections to other important classes like **hereditary algebras, tilted algebras, or self-injective algebras**.

2. Structure of Module Categories:
* **Syzygy Behavior:** Dell provides information about the eventual behavior of syzygies in the module category. Algebras with finite Dell have a certain "periodicity" in their syzygies, which could impact the structure of their **Auslander-Reiten quivers** and the **complexity of their modules**.
* **Projective Resolutions:** Dell might provide insights into the structure of minimal projective resolutions of modules. This could be particularly useful for studying **homological dimensions** and **classifying modules** up to some homological equivalence.

3. Representation-Theoretic Dichotomies:
* **Finite vs. Infinite Dell:** The distinction between algebras with finite and infinite Dell could lead to a fundamental dichotomy in representation theory. Algebras with finite Dell might have more "tame" representation theory, while those with infinite Dell could exhibit "wild" behavior.
* **Dell and Growth Properties:** Dell might be related to the **growth properties** of an algebra, such as the **polynomial growth** of its minimal projective resolutions. This could connect Dell to the **representation type** of an algebra.

4. Computational Tools:
* **Algorithmic Approaches:** Developing algorithms to compute Dell for a given algebra would be valuable. This could involve analyzing the quiver and relations of the algebra or using techniques from **computational homological algebra**.
* **Software Implementations:** Implementing Dell computations in computer algebra systems would provide a powerful tool for exploring examples, testing conjectures, and gaining a deeper understanding of this invariant.