Global Dimension of $\mathbb{Z}[C_n]$?

Question

If $C_n$ is the cyclic group of order $n$, then is there a good way to compute the global projective dimension of the integer group ring $\mathbb{Z}[C_n]$?

Answer 1

This can be easily answered if you know the characterization of global dimension by the Ext functor and the cohomology of the cyclic group. For these, see sections 4.1 and 6.2 of Weibel's book An Introduction to Homological Algebra.

Using this, we know that there are infinite indices $i$ such that $$H^i(C_n, \mathbb Z)= \mathrm{Ext}_{\mathbb Z C_n}^i(\mathbb Z,\mathbb Z)\neq 0 $$ and, therefore, the global dimension of $\mathbb Z C_n$ is infinite.

Actually, this can be proved for every non trivial finite group using this.

Answer 2

Another argument (with quite sophisticated machinery) is that if $\mathbb{Z}[C_n]$ has finite projective dimension, so has $R=\mathbb{F}_p[C_n]$ for $p|n$ (a finite exact sequence of free $\mathbb{Z}$-modules remains exact after tensoring with anything). In particular, $R$ is a regular ring finite over a field, so is reduced. But $R \supset \mathbb{F}_p[C_p] \cong \mathbb{F}_p[x]/(x-1)^p$.