Core Concepts

This paper demonstrates that while semi-valuation rings with stably coherent and regular semi-fraction rings have homotopy invariant algebraic K-theory, those with non-trivial valuations are regular but not necessarily homotopy invariant, providing examples of regular rings whose algebraic K-theory does not satisfy homotopy invariance.

Abstract

Dahlhausen, C. (2024). REGULARITY OF SEMI-VALUATION RINGS AND HOMOTOPY INVARIANCE OF ALGEBRAIC K-THEORY. arXiv preprint arXiv:2403.02413v2.

This paper investigates the relationship between the regularity of semi-valuation rings and the homotopy invariance of their algebraic K-theory. The author aims to determine if the regularity of a semi-valuation ring implies the homotopy invariance of its algebraic K-theory.

The author utilizes concepts and techniques from algebraic K-theory, commutative algebra, and algebraic geometry. They analyze the properties of semi-valuation rings, particularly their coherence and regularity, and leverage these properties to study their K-theory. The author also employs tools like Milnor squares and Temkin's relative Riemann-Zariski spaces to facilitate their analysis.

- The algebraic K-theory of semi-valuation rings with stably coherent and regular semi-fraction rings satisfies homotopy invariance.
- Semi-valuation rings with non-trivial valuations are 2-regular, meaning every finitely 2-presented module over such rings has a projective dimension of at most 1.
- Despite being regular, semi-valuation rings with non-trivial valuations are not necessarily coherent and may not have homotopy invariant algebraic K-theory.

The paper concludes that the regularity of a semi-valuation ring does not necessarily imply the homotopy invariance of its algebraic K-theory. This finding is significant as it provides a class of examples of regular rings whose algebraic K-theory does not satisfy homotopy invariance, which was an open problem in the field.

This research contributes significantly to the understanding of the relationship between the algebraic structure of rings, particularly semi-valuation rings, and their associated K-theory. It provides valuable insights into the conditions under which homotopy invariance of algebraic K-theory holds, furthering the knowledge of this fundamental invariant in algebra and topology.

The paper primarily focuses on semi-valuation rings and their K-theory. Exploring the implications of these findings for broader classes of rings and investigating other algebraic invariants in this context would be potential avenues for future research. Additionally, further research could explore the properties of relative Riemann-Zariski spaces, particularly their cohesiveness and implications for K-theory.

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by Christian Da... at **arxiv.org** 10-15-2024

Deeper Inquiries

This is a natural and important question, but the answer is not immediately clear from the paper alone. Here's why:
The Role of Regularity: The paper heavily relies on the regularity of the semi-fraction ring to establish homotopy invariance. Regularity implies that projective modules are well-behaved, which is crucial for K-theory computations. Extending to non-regular settings would require different techniques.
Milnor Squares and Descent: The proof strategy uses the fact that semi-valuation rings fit into Milnor squares, allowing for K-theory computations to be broken down. Finding analogous structures for more general rings could be a path forward.
Counterexamples: The paper provides examples of regular rings that are not K-regular (Remark 4.4). This suggests that regularity alone might not be sufficient for homotopy invariance in general, and additional conditions might be necessary.
Possible Directions for Future Research:
Weaker Notions of Regularity: Explore if weaker forms of regularity, such as those mentioned in Remark 3.4 (e.g., "weakly regular" or Gersten's "1-regular"), could be sufficient for homotopy invariance in certain cases.
Specific Classes of Rings: Focus on extending the results to specific classes of rings with properties that might be more amenable to K-theory computations, such as rings with good homological properties or those arising from geometric contexts.
Alternative Techniques: Investigate alternative approaches to proving homotopy invariance that do not rely as heavily on regularity, perhaps drawing inspiration from techniques used for other classes of rings, such as perfect Fp-algebras or C*-algebras.

This is a very interesting open question. The paper highlights the delicate interplay between regularity and homotopy invariance, particularly in the non-coherent setting.
Coherence as a Simplifying Assumption: Coherence provides a lot of control over the structure of modules, making it easier to relate different notions of regularity and to apply tools from homological algebra. In the absence of coherence, these relationships become more subtle.
Beyond Projective Dimension: The standard definitions of regularity focus on projective dimension. It's conceivable that a weaker notion could involve other homological invariants or properties that capture some aspects of regularity without requiring finite projective dimension.
Geometric Inspiration: Relative Riemann-Zariski spaces and admissible Zariski-Riemann spaces provide a geometric framework for studying semi-valuation rings. Geometric intuition might suggest alternative notions of regularity that are more naturally suited to these spaces and have implications for K-theory.
Potential Avenues for Exploration:
Homological Dimensions: Investigate if other homological dimensions, such as flat dimension or injective dimension, could play a role in a weaker notion of regularity that is still strong enough to imply homotopy invariance in some cases.
Regularity in Derived Categories: Explore notions of regularity defined in the context of derived categories, which might be more flexible and applicable to non-coherent rings.
K-Theoretic Characterizations: Seek characterizations of homotopy invariance directly in terms of K-theoretic properties, potentially bypassing the need for explicit regularity conditions.

The findings in this paper could have interesting ramifications for other algebraic invariants:
Hochschild and Cyclic Homology as Tools: Hochschild homology (HH) and cyclic homology (HC) are powerful invariants that are closely related to K-theory. They often provide more computable tools for studying rings and their modules.
Connections and Comparisons: The paper's results on the K-theory of semi-valuation rings could motivate investigations into the HH and HC of these rings. One could explore:
Explicit Computations: Can we compute HH and HC groups for specific examples of semi-valuation rings, and do these computations reflect the homotopy invariance properties observed for K-theory?
Structural Results: Are there structural theorems for HH and HC of semi-valuation rings that mirror the results for K-theory, such as excision or Mayer-Vietoris sequences?
Relative Riemann-Zariski Spaces: The geometric perspective offered by relative Riemann-Zariski spaces could provide a framework for studying the sheaf cohomology of HH and HC. This could lead to insights into the geometry of these spaces and their relationship to the underlying schemes.
Potential Research Directions:
Develop Computational Techniques: Develop techniques for computing HH and HC of semi-valuation rings, potentially leveraging the Milnor square structure or other special properties.
Explore Connections to K-Theory: Investigate the relationships between the K-theory, HH, and HC of semi-valuation rings. Are there spectral sequences or other tools that relate these invariants?
Geometric Interpretations: Seek geometric interpretations of HH and HC in the context of relative Riemann-Zariski spaces, potentially drawing connections to Hodge theory or other geometric invariants.

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