$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$

Question

Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?

These groups are also known as ideal class group in number theory.In topology,they are the home of some important obstructions due to Wall,Quinn,...(I think these groups are only known for small prime numbers $p$).

Answer 1

Let me flesh out my comment, and give some more details. I will denote $C_p$ the cyclic group of order $p$.

First, the structure of the group ring $\mathbb{Z}[C_p]$ is a little more complicated. If $\sigma$ denotes a choice of generator of $C_p$, then there is an isomorphism $\mathbb{Z}[C_p]/(\Phi_p(\sigma))\cong\mathbb{Z}[\zeta_p]$. This can be used to classify modules over $\mathbb{Z}[C_p]$, see this MO-question on representation theory over $\mathbb{Z}$ and in particular the paper of Reiner linked to there.

More relevant for the K-theory question is the fact that $\tilde{K_0}(\mathbb{Z}[C_p])$ is in fact isomorphic to the class group $\tilde{K_0}(\mathbb{Z}[\zeta_p])$ of the cyclotomic field $\mathbb{Q}(\zeta_p)$. This was shown in

• D.S. Rim. Modules over finite groups. Ann. of Math. 69 (1959), pp. 700-712,

based on the explicit description of modules over the group ring.

As a funny aside, you might want to have a look at

• M.A. Kervaire and M.P.Murthy. On the projective class group of cyclic groups or prime power order. Comment. Math. Helvetici 52 (1977), 415-452.

to see what can be said about class groups for $C_{p^n}$, $n>2$. As far as I know, these groups are not yet completely computed.

In any case, for prime order, you are then interested in the class groups of cyclotomic fields. The class numbers of cyclotomic fields for prime order roots of unity form the OEIS sequence A055513. For tables of class numbers as well as an extensive treatment on methods to understand class numbers, class groups and other facts about cyclotomic fields, see the book

• L.C. Washington. Inntroduction to cyclotomic fields, Springer, 1997 (for second edition).

For more precise statements, you better ask number theorists. As far as I understand, there are two conjectures for the $p$-part of the class group of $\mathbb{Q}(\zeta_p)$. The Kummer-Vandiver conjecture states that $p$ does not divide the +-part of the class group. A conjecture of Iwasawa states that the eigenspaces for the Galois action on the $p$-part of the class group are all cyclic groups (of order related to a special L-value). For more precise statements and references, see e.g.

• M. Kurihara. Some remarks on conjectures about cyclotomic fields and $K$-groups of $\mathbb{Z}$. Compositio Math. 81 (1992), pp. 223-236.

Summing up, an explicit list is probably only available for small primes, but there seems to be a (at least conjectural) conceptual understanding of the class groups of the group ring of cyclic groups of prime order.