# Algebraic K-theory and Homotopy Sheaves

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?

-

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").