# Group rings and projective modules

If $A$ is a lattice (i.e. a fin. gen. free $\mathbb{Z}$-module), and $G$ is some group which acts on $A$, will $A$ be a projective $\mathbb{Z}[G]$-module?

Thanks

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## 1 Answer

Clearly not, as the action can be trivial!

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And it is very rare that $\mathbb Z$ be a projective $\mathbb Z[G]$-module. – Mariano Suárez-Alvarez Dec 20 '12 at 16:30
Ah yes I didnt think about the action being trivial, thank you. – Chris Birkbeck Dec 20 '12 at 16:42
Notice that there are many other actions which will give non-projective modules, also. – Mariano Suárez-Alvarez Dec 20 '12 at 16:56