K-groups of a field
In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen.
Low degrees
The map sending a finite-dimensional F-vector space to its dimension induces an isomorphism
- [math]\displaystyle{ K_0(F) \cong \mathbf Z }[/math]
for any field F. Next,
- [math]\displaystyle{ K_1(F) = F^\times, }[/math]
the multiplicative group of F.[1] The second K-group of a field is described in terms of generators and relations by Matsumoto's theorem.
Finite fields
The K-groups of finite fields are one of the few cases where the K-theory is known completely:[2] for [math]\displaystyle{ n \ge 1 }[/math],
- [math]\displaystyle{ K_n(\mathbb{F}_q) = \pi_n(BGL(\mathbb{F}_q)^+) \simeq \begin{cases} \mathbb{Z}/{(q^i - 1)}, & \text{if }n = 2i - 1 \\ 0, & \text{if }n\text{ is even} \end{cases} }[/math]
For n=2, this can be seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture. A different proof was given by (Jardine 1993).
Local and global fields
(Weibel 2005) surveys the computations of K-theory of global fields (such as number fields and function fields), as well as local fields (such as p-adic numbers).
Algebraically closed fields
(Suslin 1983) showed that the torsion in K-theory is insensitive to extensions of algebraically closed fields. This statement is known as Suslin rigidity.
See also
- divisor class group
References
- ↑ Weibel 2013, Ch. III, Example 1.1.2.
- ↑ Weibel 2013, Ch. IV, Corollary 1.13.
- Jardine, J. F. (1993), "The K-theory of finite fields, revisited", K-Theory 7 (6): 579–595, doi:10.1007/BF00961219
- Suslin, Andrei (1983), "On the K-theory of algebraically closed fields", Inventiones Mathematicae 73 (2): 241—245, doi:10.1007/BF01394024
- Weibel, Charles (2005), "Algebraic K-Theory of Rings of Integers in Local and Global Fields", in Friedlander, Eric M.; Grayson, Daniel R., Handbook of K-Theory, Springer, pp. 139—190, doi:10.1007/978-3-540-27855-9_5, ISBN 978-3-540-27855-9
- Weibel, Charles A. (2013), The K-book, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, ISBN 978-0-8218-9132-2, http://www.math.rutgers.edu/~weibel/Kbook.html
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