Gabber rigidity theorem

Remarks

Gabber proved the theorem in the more general setting of Henselian pairs $(A, I)$, comparing $A$ with $A/I$. The proof develops the specialization argument of Suslin's rigidity theorem, with the Henselian property used to lift points from the residue field to the ring.

At work: the ring of $p$-adic integers $\Z_p$ is Henselian with residue field $\F_p$. For $\ell$ prime to $p$, the theorem gives $K_n(\Z_p; \Z/\ell) \cong K_n(\F_p; \Z/\ell)$, known from Quillen's computation for finite fields.

The hypothesis that $\ell$ is invertible is necessary. Let $p$ be odd and $\ell = p$. Then $K_1(\Z_p; \Z/p) = \Z_p^\times/(\Z_p^\times)^p \cong \Z/p$, since $\Z_p^\times \cong \mu_{p-1} \times (1 + p\Z_p)$ and $1 + p\Z_p \cong \Z_p$. But $K_1(\F_p; \Z/p) = \F_p^\times/(\F_p^\times)^p = 0$, because $\F_p^\times$ has order $p - 1$, prime to $p$.

Going deeper

For a student

The theorem computes the stalks of mod $\ell$ K-theory in the étale topology. The local rings there are strictly Henselian, so their mod $\ell$ K-groups are those of separably closed fields. This is the starting point of the comparison between algebraic K-theory and étale cohomology, in étale K-theory and the Quillen–Lichtenbaum conjecture. It extends Suslin's rigidity theorem for extensions of algebraically closed fields.