# Tensor Product of Witt Vectors

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.

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.
• Maybe it's also worth pointing out that without the etaleness assumption, it's usually false (but not always). For example, it's false for $C=\mathbf{F}_p$, $A=B=C[x]$, $n=2$. Also, in the case where $C$ is an $\mathbf{F}_p$-algebra, the result is due to Illusie, in his ENS paper on the de Rham-Witt complex. Mar 4 '11 at 12:41