K-groups of strict henselization of stalks

Question

How well are the algebraic K-groups of the strict henselization of the stalks $\mathcal{O}_{X,p}^{sh}$ at geometric points of a scheme $X$ understood? I am particularly interested in the case of rational K-theory and when $X$ is a smooth projective scheme over $\mathbb{F}_p$.

Answer 1

As remarked in the comment of Donu Arapura, the K-theory with finite coefficients (or even completed at some prime) can be understood using the rigidity theorems of Suslin and Gabber. These state that the projection $\mathcal{O}_{X,x}^{sh}\to\kappa(x)$ from the strictly henselian local ring to its residue field induces an isomorphism on K-theory with finite coefficients. This reduces the question to algebraically closed fields in characteristic $p$, but the rigidity theorem also tells us that inclusions of algebraically closed fields induce isomorphisms on K-theory with finite coefficients. With finite coefficients, we thus know a complete answer (because we know the K-theory of $\overline{\mathbb{F}_q}$ from Quillen's computations).

With rational coefficients, however, K-theory is not understood, as far as I know. The rigidity theorems are no longer true (which can already be seen on $K_1$). Already the special case of the K-theory of algebraic closures of function fields of varieties in characteristic $p$ is unclear; we don't know $K_\bullet(\overline{\mathbb{F}_q(T_1,\dots,T_n)})\otimes\mathbb{Q}$ for $n>2$. For $n=1$, we can approximate $K_\bullet(\overline{\mathbb{F}_q(T)})$ as a filtered colimit of $K_\bullet(\mathbb{F}_q[C_i])$ where $C_i$ runs through smooth curve models of finite extensions of $\mathbb{F}_q(T)$. Then by the work of Harder, we know that the K-groups $K_i(\mathbb{F}_q[C])$ of these curves are torsion for $i>1$. Passing to the colimit, we see that the rational K-groups of $\overline{\mathbb{F}_q(T)}$ vanish in degrees $>1$. In degree 1, we just have $\overline{\mathbb{F}_q(T)}^\times\otimes_{\mathbb{Z}}\mathbb{Q}$.

There is also Parshin's conjecture on vanishing of K-theory, but that concerns smooth projective varieties, so it doesn't immediately say much about the K-theory of function fields of varieties. Maybe one could use a localization argument to inductively deduce from Parshin's conjecture and the curve case that the rational K-theory of $\overline{\mathbb{F}_q(T_1,\dots,T_n)}$ vanishes in degrees $>n$, but I haven't checked that properly.