Let $A$, $B$, and $C$ be commutative rings such that $A\otimes\_C B$ makes sense. If $W\_n(A\otimes\_C B), W\_n(A), W\_n(C),$ and $W\_n(B)$ are the length $n$ Witt vectors of the rings $A,B,C,$ and $A\otimes\_C B$. Is it true that 


$$
W\_n(A\otimes\_C B)\cong W\_n(A)\otimes\_{W\_n(C)}W\_n(B)?
$$


It seems as though this is a sensible property for Witt vectors to have. The case I am particularly interested in is the case when $C$ is a field of characteristic $p$ (not necessarily perfect) and $A$ and $B$ are $C$-algebras, but any suggestions for the general case would be helpful as well.



Tensor product of rings of Witt vectors
When $B$ is étale over $C$ and $A$ or $B$ is finite over $C$,
then the result is known by Theorem (2.4) in my paper
[Descent for the $K$-theory of polynomial rings](http://www.digizeitschriften.de/id/266833020_0191%7Clog39?tify=%7B%22pages%22:%5B412,413%5D,%22view%22:%22%22%7D&origin=/search?q%3Ddescent%2520for%2520the%2520k-theory).


