I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy theory. On the other hand, I've been warned about the fact that algebraic $K$-theory isn't always $\mathbb A^1$-local. By "algebraic $K$-theory,", I mean the algebraic $K$-theory of perfect complexes of quasicoherent sheaves (I think -- let me know if I should mean something else).


* I'm pretty sure that Thomason and Trobaugh show that algebraic $K$-theory  always  (in the quasicompact, quasiseparated case) satisfies Nisnevich descent.
* Under certain conditions, algebraic $K$-theory is $\mathbb A^1$-local and has some kind of compatibility with $\mathbb G\_m$ which should make it $\mathbb P^1$-local. I think  Weibel  (already Quillen) calls this "the fundamental theorem of algebraic $K$-theory".


So putting this together, let $S$ be a scheme, and let $SH(S)$ be the stable motivic ($\infty$-)category over $S$.


**Questions:**


1. Is algebraic $K$-theory of smooth schemes over $S$ representable as an object of $SH(S)$?
2. How about if we put some conditions on $S$ -- say it's regular, noetherian, affine, smooth over an algebraically closed field? Heck, what if we specialize to $S = Spec(\mathbb C)$?
3. Does it make a difference if we redefine $SH(S)$ to be certain sheaves of spectra over the site of smooth schemes affine over $S$ or something like that?



Is algebraic $K$-theory a motivic spectrum?
Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL\_\infty\times\mathbb{Z}$ in each level (so it is a $\mathbb{P}^1$-periodic motivic spectrum).


This is theorem 4.3.13 in



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> *Morel, Fabien; Voevodsky, Vladimir*, [**$\bf A^1$-homotopy theory of schemes**](http://dx.doi.org/10.1007/BF02698831), Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999). [ZBL0983.14007](https://zbmath.org/?q=an:0983.14007).
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A correct proof of the above result can be found in [this survey](https://arxiv.org/abs/1605.00929) (thanks to Marc Hoyois for pointing out that the original proof was incorrect).


The proof of the fact that algebraic K-theory is a motivic spectrum goes exactly how you described:


* It satisfies Nisnevich descent on qcqs schemes by Thomason-Trobaugh's main theorem.
* It is $\mathbb{A}^1$-invariant on *regular Noetherian schemes* (which every scheme smooth over $S$ is ) by Quillen's fundamental theorem of K-theory
* It is $\mathbb{P}^1$-periodic by the projective bundle formula (also in Thomason-Trobaugh)


When $S$ is not regular noetherian the situation is more complicated. You can still represent $K$ by some version of the infinite Grassmannian (proposition 4.3.14 in Morel-Voevodsky), but this object won't be $\mathbb{A}^1$-invariant anymore. What you can consider is the homotopy K-theory presheaf $KH=Sing\_\* K$. This object is indeed represented by $BGL\_\infty\times \mathbb{Z}$. This result has been announced in Voevodsky's ICM address and proven in



> 
> *Cisinski, Denis-Charles*, [**Descent by blow-ups for homotopy invariant $K$-theory**](http://dx.doi.org/10.4007/annals.2013.177.2.2), Ann. Math. (2) 177, No. 2, 425-448 (2013). [ZBL1264.19003](https://zbmath.org/?q=an:1264.19003).
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(thanks to Marc Hoyois for the reference to this result)


