# 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?

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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

The question already has good answers but I think there is still more to be said.

### References

As already mentioned, algebraic K-theory satisfies Zariski descent. For regular noetherian schemes this is due to

• Kenneth S. Brown, Stephen M. Gersten, Algebraic K-theory as generalized sheaf cohomology, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.

It was generalized to finite dimensional noetherian schemes by

• R. Thomason, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, vol. III, Progress in Mathematics, vol. 88, Birkhäuser, Basel, 1990, pp. 247–435.

However, there is a stronger statement: it satisfies descent with respect to the Nisnevich topology (which lies between Zariski and etale). This is due to

• Yevsey A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.

and was generalized to finite dimensional noetherian schemes in the same paper of Thomason.

In the following paper the above results are extended to finite dimensional quasi-compact quasi-separated schemes.

• Andreas Rosenschon, P.A. Ostvær, Descent for K-theories, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.

### Sketch of a proof

I learned from Peter Scholze that in modern language a proof can be given, after identifying Quillen K-theory with the K-theory of the stable infinity-category of perfect complexes, by using the following characterization of extendibility of perfect complexes: a perfect complex on an open subscheme $U \subset X$ can be extended up to quasi-isomorphism to a perfect complex on $X$, if and only if its class in $K_0(U)$ lies in the image of $K_0(X)$. This kind of thing is discussed in

• Bhargav Bhatt, Algebraization and Tannaka duality, 2014, arXiv.

This gives a homotopy fibre sequence of connective spectra $K(X \text{ on } Z) \to K(X) \to K(U)$ where $Z \subset X$ is the closed complement of an open subscheme $U \subset X$. I am not sure in exactly what generality this proof works. The significance of working with infinity-categories is that the presheaf of infinity-categories $U \mapsto \mathrm{Perf}(U)$ satisfies descent, unlike its triangulated shadow.

### Etale descent

Finally, let me mention that the question of etale descent is closely related to the Lichtenbaum-Quillen conjecture. This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees. The theorem of Trobaugh that Steven Landsburg mentioned, about etale descent for mod-$\ell$ Bousfield-localized K-theory, is in

• R. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.

and is also generalized in the paper of Rosenschon and Ostvær.

### Sketch of a proof, part 2

Let me give more details on the proof, now that I understand the details a little better. The main ingredients are

1. Nisnevich descent for perfect complexes, as a prestack of stable infinity-categories $X \mapsto Perf(X)$.

2. Compact generation of $D(X)$ by the perfect complexes, and also for the version "with support" on a closed subscheme $Z \subset X$, i.e. for the full subcategory $D_Z(X)$ of complexes that vanish on $X-Z$.

3. The Thomason-Neeman localization theorem: an exact sequence of stable infinity-categories induces a fibre sequence of K-theory spectra.

Then the proof is as follows. Let $j : U \hookrightarrow X$ be an open immersion and $p : Y \to X$ an etale morphism defining an elementary Nisnevich square

$$\require{AMScd} \begin{CD} W @>>> Y \\ @VVV @VV{p}V \\ U @>>{j}> X \end{CD}$$

We want to show that the square

$$\require{AMScd} \begin{CD} K(X) @>{j^*}>> K(U) \\ @V{p^*}VV @VVV \\ K(Y) @>>> K(W) \end{CD}$$

is a homotopy cartesian square of connective spectra. (In the noetherian case, Morel-Voevodsky showed that Nisnevich descent is equivalent to this Brown-Gersten-style excision property; in the non-noetherian case, this should really be taken as the "correct" definition of the Nisnevich topology.) For this it is sufficient to show that there is an equivalence on homotopy fibres. 2) and 3) imply that the homotopy fibres are given by $K_{X-U}(X)$ and $K_{Y-W}(Y)$, respectively. Then the equivalence follows from the fact that there is already an equivalence

$$Perf_{X-U}(X) \stackrel{\sim}{\longrightarrow} Perf_{Y-W}(Y)$$

at the level of perfect complexes, by 1).

This proof works for quasi-compact quasi-separated schemes. By the way, the same proof works for (qcqs) derived schemes. By far the most non-trivial part of the proof is 2), which was established in [B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599]. A proof of 1), attributed to Drinfeld, is given in [D. Gaitsgory, Notes on geometric Langlands: Quasi-coherent sheaves on stacks, pdf]. In fact, this probably even gives a proof for the spectral schemes of Lurie, modulo a key point in the proof of 2) which I do not know how to do in the setting of $E_\infty$-ring spectra (but this is probably just my ignorance).

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Adeel, could you elaborate on your statement of extensions of perfect complexes? I mean, you seem to suggest that the obstruction to, say, a vector bundle extending from U to all of X lies in just $K_0(U)$? Is that true? I've never heard of this statement, so I'd like to know more. –  bananastack Sep 6 '14 at 23:44
@user125763 Lemma 5.5.1 in Thomason-Trobaugh says that for any perfect complex $F$ on $U$, there is a perfect complex $E$ on $X$ such that $F$ is isomorphic to a direct summand of $j^*E$ in $D^-(\mathcal{O}_U-Mod)$. This is, apparently, a result of Trobaugh's posthumous simulacrum appearing in Thomason's dream. –  S. Carnahan Sep 7 '14 at 0:01
thanks @S.Carnahan: is that only apparently weaker than what Adeel claimed (by some manipulations of summands)? Or is it actually weaker? –  bananastack Sep 7 '14 at 0:12
I guess it's an intermediate step to the main result (Theorem 7.4). See Clark Barwick's answer: mathoverflow.net/questions/5580/… –  S. Carnahan Sep 7 '14 at 0:31
Thanks, Adeel, this is good. Have added these references here: ncatlab.org/nlab/show/algebraic+K-theory#Descent –  Urs Schreiber Sep 7 '14 at 10:57

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

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There is a nice modern account of the Zariski-descent statement and nice perspective on the resulting sheaf of algebraic K-theory $E_\infty$-rings on the arithmetic site in

• Ulrich Bunke, Georg Tamme, section 3.3 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

and for the $E_\infty$-multiplicative refinement in

• Ulrich Bunke, Georg Tamme, section 2.4 of Multiplicative differential algebraic K-theory and applications (arXiv:1311.1421)

They go further and extend this sheaf of algebraic K-theory spectra to the site of products of arithmetic schemes with a smooth manifold and produce there a differential refinement to a sheaf of differential algebraic K-theory-spectra which naturally supports the Beilinson regulator maps as a single map of sheaves of spectra.

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