Is algebraic $K$-theory a motivic spectrum? 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).


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*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:


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*Is algebraic $K$-theory of smooth schemes over $S$ representable as an object of $SH(S)$?

*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 $S = Spec(\mathbb C)$?

*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?
 A: 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

Morel, Fabien; Voevodsky, Vladimir, $\bf A^1$-homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999). ZBL0983.14007.

A correct proof of the above result can be found in this survey (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:


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*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, Ann. Math. (2) 177, No. 2, 425-448 (2013). ZBL1264.19003.

(thanks to Marc Hoyois for the reference to this result)
