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 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
https://arxiv.org/pdf/2403.02413.pdfDeeper Inquiries