It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong K_\ast(k,\mathbb{Z}/l)$. My question is: are there any more classes of (regular) local rings such that this is true for them? Conversely, for which types of local rings this statement is 'usually' wrong?
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asked Jan 11, 2013 at 18:54
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Yes, that is a theorem of Gabber. I recommend the survey by Grayson.
I will suggest how to create other examples of such rings, but they should be considered accidents. Later, Nisnevich clarified the theorem by proving that $K$-theory is a homotopy sheaf in the Nisnevich topology, thus interpreting Gabber's result as identifying the stalks of this sheaf. Roughly speaking, the sheaf condition tells you how to compute the $K$-theory of the ring from that of its henselization. In particular, they are unlikely to be the same and thus they should not both match the residue field.
In the one-dimensional case, the discrepancy between the $K$-theories of a ring and its henselization is the same as the discrepancy between the $K$-theories of their fraction fields. In fact, $K$-theory is not just a Nisnevich sheaf, but almost an étale sheaf. In particular, Thomason proved that if you invert the Bott element, mod $l$ $K$-theory becomes étale $K$-theory, which is approximately étale cohomology. So if the fraction field of the ring and the fraction field of the henselization have the same mod $l$ $K$-theories, they have the same mod $l$ cohomologies, which is rather rare. For a start, the index of the Galois groups should be prime to $l$; that is, the ring should satisfy the hensel property for degree $l$ extensions. This is not enough, but if both absolute Galois groups are prime to $l$, then they have no mod $l$ cohomology, thus fairly simply and identical étale $K$-theories; and probably they have the same mod $l$ algebraic $K$-theories.
