Rouquier Dimension Does Not Imply Finite Global Dimension, But It Does Imply Finite Weak Global Dimension for Coherent Rings

Kernkonzepte

Contrary to a previous claim, a ring having finite Rouquier dimension for its category of perfect complexes does not guarantee finite global dimension. However, for coherent rings, it is equivalent to having finite weak global dimension.

Zusammenfassung

Stevenson, G. (2024). Rouquier dimension versus global dimension. arXiv preprint arXiv:2410.04793v1.

This research paper disproves a claim made in prior work, stating that finite Rouquier dimension of the category of perfect complexes over a commutative ring implies finite global dimension. The paper aims to clarify the relationship between these concepts and provide a corrected statement.

Wichtige Erkenntnisse aus

Rouquier dimension versus global dimension

by Greg Stevens... um arxiv.org 10-08-2024

https://arxiv.org/pdf/2410.04793.pdf

Tiefere Fragen

How does the concept of Rouquier dimension relate to other homological invariants in more general categorical settings beyond module categories?

Rouquier dimension, while often discussed in the context of module categories, is fundamentally a property of triangulated categories. This makes it applicable to a much broader range of settings beyond just modules over rings. Here's how it connects to other homological invariants in this broader context:

Generalization of Global Dimension: In the realm of abelian categories (like module categories), global dimension measures the resolvability of objects by projective objects. Rouquier dimension can be seen as a more flexible analogue for triangulated categories. While it doesn't directly measure projective resolutions, it captures how efficiently objects can be built from a given generating set using shifts, cones, and direct summands.

Relationship to Generation Time: Rouquier dimension provides a quantitative measure for the "generation time" of a triangulated category. A low Rouquier dimension implies that every object can be built relatively quickly from the generators. This connects to concepts like the Orlov spectrum of a category, which studies the minimal length of semiorthogonal decompositions.

Applications in Algebraic Geometry and Topology: Triangulated categories are fundamental in algebraic geometry (derived categories of sheaves) and topology (stable homotopy category). In these settings, Rouquier dimension provides insights into the complexity of the category and its relationship to the underlying geometric or topological space. For example, it can be used to study the derived categories of singular varieties.

Connections to K-Theory: Rouquier dimension has implications for the K-theory of triangulated categories. Finite Rouquier dimension often implies regularity properties in K-theory, which are important for understanding the structure of the category.
In summary, Rouquier dimension acts as a bridge between the abstract world of triangulated categories and concrete homological properties. Its connections to generation time, global dimension (in special cases), and K-theory make it a powerful tool for studying a wide range of mathematical objects.

Could there be a weaker condition than coherence that still implies a relationship between the Rouquier dimension of the perfect complexes and some variant of global dimension?

Yes, it's plausible that weaker conditions than coherence could still yield a relationship between the Rouquier dimension of perfect complexes and some variant of global dimension. Here are some avenues to explore:

Semi-Coherence: A ring is semi-coherent if every finitely generated submodule of a finitely presented module is finitely presented. This condition is weaker than coherence. It's possible that finite Rouquier dimension of Dperf(R) for a semi-coherent ring R could imply a bound on the finitistic projective dimension of R, which is the supremum of the projective dimensions of finitely generated modules with finite projective dimension.

Locally Coherent: Instead of requiring the entire ring to be coherent, one could consider rings that are locally coherent, meaning that every localization at a prime ideal is coherent. This could potentially lead to a relationship between the Rouquier dimension of Dperf(R) and a "local" version of global dimension, perhaps defined in terms of the global dimensions of localizations of R.

Weaker Homological Conditions: Instead of focusing on projective or flat dimensions, one could explore connections to other homological invariants that might be better behaved in the absence of coherence. For example, the Gorenstein projective dimension or the completely presented dimension might offer alternative perspectives.

Restrictions on the Derived Category: Imposing additional conditions on the structure of Dperf(R), such as the existence of certain adjoint functors or bounds on the thicknesses of certain subcategories, might lead to relationships with weaker notions of global dimension.
Investigating these weaker conditions would require a careful analysis of how the proofs connecting Rouquier dimension and global dimension break down in the absence of coherence. It's a challenging but potentially fruitful direction for research.

What are the implications of this corrected understanding of Rouquier dimension for the development of non-commutative geometry, where geometric concepts are studied using categorical and homological tools?

The corrected understanding of Rouquier dimension, particularly its nuanced relationship with global dimension in non-Noetherian settings, has significant implications for non-commutative geometry:

Refined Geometric Invariants: Non-commutative geometry seeks to extend geometric notions to settings where the "space" is represented by a non-commutative ring or, more generally, a category. Rouquier dimension, being a categorical invariant, provides a tool to probe the "geometric complexity" of these non-commutative spaces. The realization that it doesn't directly correspond to global dimension in all cases necessitates a more refined understanding of how different homological dimensions capture geometric properties.

Focus on Weaker Notions of Dimension: The example presented in the paper highlights the importance of considering weaker notions of dimension, such as weak global dimension or finitistic dimensions, when studying non-commutative spaces. These weaker notions might provide a more accurate reflection of the geometric intuition in situations where classical global dimension is poorly behaved.

Importance of Coherence: The paper underscores the crucial role of coherence in connecting Rouquier dimension to global dimension. This emphasizes the need to carefully consider the role of finiteness conditions in non-commutative geometry. It suggests that different classes of non-commutative rings (coherent, semi-coherent, etc.) might require distinct geometric interpretations.

Derived Categories as Geometric Tools: The use of derived categories and their invariants, like Rouquier dimension, is becoming increasingly central in non-commutative geometry. This corrected understanding encourages a more cautious and nuanced approach to using derived categories as geometric tools, paying close attention to the interplay between categorical properties and geometric intuition.

New Avenues for Research: This development opens up new research directions. For instance, exploring the geometric meaning of Rouquier dimension for non-coherent rings or investigating the relationship between Rouquier dimension and other homological dimensions in a non-commutative setting could lead to a deeper understanding of non-commutative spaces.
In conclusion, the corrected understanding of Rouquier dimension prompts a reevaluation of how we use homological tools to study non-commutative spaces. It encourages a more refined and cautious approach, paying close attention to the limitations of classical invariants in non-commutative settings and exploring alternative notions of dimension that better capture the geometry.