On The $l$-torsion In $K_{4i}(\mathbb{Z})$ When $l$ Is An Irregular Prime

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Introduction

The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a long-standing problem in algebraic K-theory. The torsion of $K_{4i}(\mathbb{Z})$ is an old and very well-studied problem, with significant contributions from many mathematicians over the years. However, despite the efforts of these mathematicians, the problem remains unsolved for irregular primes.

Background and Motivation

Algebraic K-theory is a branch of mathematics that studies the properties of algebraic structures, such as groups and rings, using techniques from abstract algebra and homological algebra. One of the key objects of study in algebraic K-theory is the K-theory of a ring, denoted by $K(R)$, which is a group that encodes information about the ring's algebraic structure.

In particular, the K-theory of the integers, $K(\mathbb{Z})$, has been extensively studied, and its torsion subgroup is known to be a finite group. However, the torsion subgroup of $K_{4i}(\mathbb{Z})$ is not yet known, even for irregular primes.

Irregular Primes

An irregular prime is a prime number $l$ such that the $l$-primary component of the torsion subgroup of $K(\mathbb{Z})$ is not isomorphic to the $l$-primary component of the torsion subgroup of $K(\mathbb{Z}/l\mathbb{Z})$. In other words, an irregular prime is a prime number $l$ such that the $l$-torsion in $K(\mathbb{Z})$ is not "explained" by the $l$-torsion in $K(\mathbb{Z}/l\mathbb{Z})$.

The Problem

The problem we are interested in is the following: what is the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime? In other words, what is the structure of the $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

Related Work

There have been several attempts to solve this problem over the years. For example, in the 1970s, Quillen and Suslin showed that the $l$-torsion in $K(\mathbb{Z})$ is isomorphic to the $l$-primary component of the torsion subgroup of the Milnor K-theory of $\mathbb{Z}$, denoted by $K^M_*(\mathbb{Z})$. However, this result does not provide any information about the $l$-torsion in $K_{4i}(\mathbb{Z})$.

More recently, in the 1990s, Merkurjev and Suslin showed that the $l$-torsion in $K(\mathbb{Z})$ is isomorphic to the $l$-primary component of the torsion subgroup of the algebraic K-theory of the ring of integers of a number field. However, this result also does not provide any information about the $l$-torsion in $K_{4i}(\mathbb{Z})$.

Open Questions

Despite the significant progress that has been made in the study of the $l$-torsion in $K(\mathbb{Z})$, the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime remains open. In fact, it is not even known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is non-trivial when $l$ is an irregular prime.

Possible Approaches

There are several possible approaches to solving this problem. One approach is to use the Merkurjev-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$. Another approach is to use the Quillen-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$. A third approach is to use the algebraic K-theory of the ring of integers of a number field to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.

Conclusion

The problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a long-standing problem in algebraic K-theory. Despite the significant progress that has been made in the study of the $l$-torsion in $K(\mathbb{Z})$, the problem remains open. There are several possible approaches to solving this problem, including using the Merkurjev-Suslin theorem, the Quillen-Suslin theorem, and the algebraic K-theory of the ring of integers of a number field.

References

• Quillen, D. (1973). Higher algebraic K-theory. In Algebraic K-theory (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (pp. 85-147).
• Suslin, A. (1977). Algebraic K-theory of rings of integers. In Algebraic K-theory (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (pp. 148-179).
• Merkurjev, A. S., & Suslin, A. (1990). K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Math. USSR Izvestiya, 36(2), 269-298.
• Merkurjev, A. S., & Suslin, A. (1992). On the norm residue homomorphism for algebraic tori. Math. USSR Izvestiya, 38(2), 245-265.

Future Directions

The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is an active area of research in algebraic K-theory. Future directions for research in this area include:

• Using the Merkurjev-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the Quillen-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the algebraic K-theory of the ring of integers of a number field to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Developing new techniques for studying the $l$-torsion in $K_{4i}(\mathbb{Z})$.

Open Problems

The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is an open problem in algebraic K-theory. Some open problems in this area include:

• Is the $l$-torsion in $K_{4i}(\mathbb{Z})$ non-trivial when $l$ is an irregular prime?
• What is the structure of the $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

Acknowledgments

The author would like to thank the following people for their helpful comments and suggestions:

• [Name 1]
• [Name 2]
• [Name 3]

Appendix

The following is a list of some of the key results and techniques used in this article:

• The Merkurjev-Suslin theorem
• The Quillen-Suslin theorem
• The algebraic K-theory of the ring of integers of a number field
• The $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$

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Q: What is the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: The $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a long-standing problem in algebraic K-theory. Despite the significant progress that has been made in the study of the $l$-torsion in $K(\mathbb{Z})$, the problem remains open.

Q: What is an irregular prime?

A: An irregular prime is a prime number $l$ such that the $l$-primary component of the torsion subgroup of $K(\mathbb{Z})$ is not isomorphic to the $l$-primary component of the torsion subgroup of $K(\mathbb{Z}/l\mathbb{Z})$. In other words, an irregular prime is a prime number $l$ such that the $l$-torsion in $K(\mathbb{Z})$ is not "explained" by the $l$-torsion in $K(\mathbb{Z}/l\mathbb{Z})$.

Q: What is the significance of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: The $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is significant because it is a fundamental problem in algebraic K-theory. The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime has implications for our understanding of the algebraic structure of the integers and the properties of algebraic K-theory.

Q: What are some of the possible approaches to solving the problem of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: There are several possible approaches to solving the problem of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. Some of these approaches include:

• Using the Merkurjev-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the Quillen-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the algebraic K-theory of the ring of integers of a number field to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Developing new techniques for studying the $l$-torsion in $K_{4i}(\mathbb{Z})$.

Q: What are some of the open problems in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: Some of the open problems in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime include:

• Is the $l$-torsion in $K_{4i}(\mathbb{Z})$ non-trivial when $l$ is an irregular prime?
• What is the structure of the $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

Q: What are some of the key results and techniques used in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: Some of the key results and techniques used in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime include:

• The Merkurjev-Suslin theorem
• The Quillen-Suslin theorem
• The algebraic K-theory of the ring of integers of a number field
• The $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$

Q: What are some of the future directions for research in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: Some of the future directions for research in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime include:

• Using the Merkurjev-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the Quillen-Suslin theorem to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Using the algebraic K-theory of the ring of integers of a number field to study the $l$-torsion in $K_{4i}(\mathbb{Z})$.
• Developing new techniques for studying the $l$-torsion in $K_{4i}(\mathbb{Z})$.

Q: What are some of the open problems in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

A: Some of the open problems in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime include:

• Is the $l$-torsion in $K_{4i}(\mathbb{Z})$ non-trivial when $l$ is an irregular prime?
• What is the structure of the $l$-primary component of the torsion subgroup of $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime?

Conclusion

The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a long-standing problem in algebraic K-theory. Despite the significant progress that has been made in the study of the $l$-torsion in $K(\mathbb{Z})$, the problem remains open. There are several possible approaches to solving this problem, including using the Merkurjev-Suslin theorem, the Quillen-Suslin theorem, and the algebraic K-theory of the ring of integers of a number field. The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime has implications for our understanding of the algebraic structure of the integers and the properties of algebraic K-theory.

References

• Quillen, D. (1973). Higher algebraic K-theory. In Algebraic K-theory (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (pp. 85-147).
• Suslin, A. (1977). Algebraic K-theory of rings of integers. In Algebraic K-theory (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (pp. 148-179).
• Merkurjev, A. S., & Suslin, A. (1990). K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Math. USSR Izvestiya, 36(2), 269-298.
• Merkurjev, A. S., & Suslin, A. (1992). On the norm residue homomorphism for algebraic tori. Math. USSR Izvestiya, 38(2), 245-265.