Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat $R$-modules. Then does $H(C\otimes\_R S)\cong H(C)\otimes\_R S$ whenever $H(C)$ is flat $R$-module. $H(C)$ is homology of chain complex $C$.$S$ is commutative ring over $R$


I checked wikipedia and nLab,it seems that there is only the case when $R$ is field or Principal ideal domain.I dont know whether one can use so called Kunneth spectral sequence to make arguments.I just started to understand


Thanks 



Universal coefficient theorem for local ring
It doesn't hold. Take $k$ any field (of characteristic zero if you wish), $R=k[\epsilon]/(\epsilon^2)$ its ring of dual numbers, $S=R/(\epsilon)=k$ its residue field, and $C$ the periodic complex
$$\cdots\rightarrow R\stackrel{\epsilon}\longrightarrow R\rightarrow\cdots.$$
This is a complex of projective (hence flat) $R$-modules with trivial (hence flat) homology $H(C)=0$, so $H(C)\otimes\_RS=0$, but $C\otimes\_RS$ is 
$$\cdots\rightarrow S\stackrel{0}\longrightarrow S\rightarrow\cdots,$$
hence $H\_n(C\otimes\_RS)=S\neq 0$ in each degree $n\in\mathbb Z$.


