Is [Gabriel's theorem](https://en.wikipedia.org/wiki/Gabriel%27s_theorem) on the indecomposables of representations of quivers of finite type true over a commutative ring, i.e. not necessarily a field?



Gabriel's theorem over a commutative ring
Certainly it doesn't generalize in a really obvious way. One way to think about this is the following: Gabriel's theorem uses in really deep way that the category of quiver representations over a field is hereditary (in particular, $\mathrm{Ext}^2(M,N)=0$ for any representations). Over an arbitrary commutative ring, this is false, and so lots of things go out the window. For example, my preferred proof of Gabriel's theorem is from Crawley-Boevey's notes (<http://www1.maths.leeds.ac.uk/~pmtwc/quivlecs.pdf>); perhaps the most important lemma there is that a Dynkin quiver can't have any indecomposable quiver representations that have automorphisms other than scalars, or any self-extensions. Of course, this is totally false over a non-semisimple ring.


