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Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory provide us with a $\mathrm{K}$-theory space. Certain constructions also give us explicit deloopings, and if the ring is commutative we even get an $E_\infty$ structure on the resulting spectrum. Now, all of a sudden, we have a sheaf of $E_\infty$-rings over the Zariski site of a ring $R$: namely, for the affine open $\mathrm{Spec}(R_f)$ of $\mathrm{Spec}(R)$, we assign the $\mathrm{K}$-theory spectrum $\mathcal{K}(R_f)$. We can, of course, do this over the etale site as well. My questions are:

  1. Does this assignment satisfy Zariski, etale descent?

  2. If not, what type of modifications can we do for it to satisfy the above-mentioned descents?

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2 Answers 2

up vote 10 down vote accepted

For the Zariski topology, one has cohomological descent if $R$ is regular. (This yields the Brown-Gersten spectral sequence.)

For the etale topology, still assuming $R$ regular, descent fails for $K$-theory but one has descent for the theory $K/(p^\nu)[\beta^{-1}]$ where $p^\nu$ is a prime power and $\beta$ is the Bott element. This is a theorem of Thomason (see his paper "Algebraic K-Theory and Etale Cohomology").

Thomason also showed in a later paper that in the etale case, you can replace the regularity assumption with some some technical assumptions including finite Krull dimension and some bounds on the etale cohomological dimension of the residue fields.

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Thanks for the answer. –  nerses Jun 1 '13 at 4:15

It seems to me that this paper answers positively to your question.

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