9
$\begingroup$

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.

$\endgroup$

1 Answer 1

14
$\begingroup$

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.

$\endgroup$
1
  • 15
    $\begingroup$ 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. $\endgroup$
    – JBorger
    Mar 4, 2011 at 12:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.