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 $C$.

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

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