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I have a few naive questions on the algebraic K-theory spectrum construction, but whose answers I couldn't figure out using the internet. I'm mostly interested in the case of a commutative ring, but I guess most of these questions apply somewhat more generally as well - e.g. $E_{\infty}$ ring spectra, or maybe more abstractly for something like exact monoidal categories.

1) I've seen several ways to construct the cup product in algebraic K-theory, which generally result in something on the level of the K-space, i.e. a map $K\wedge K\to K$ which is associative, commutative up to homotopy and distributive over the H-space structure up to homotopy. Am I correct that this also implies the corresponding connective spectrum is a ring spectrum?

2) Under what circumstances do the above structures lift to $\mathbb{E}_\infty$-ring structures? I would guess if you had similar coherence structures already on the symmetric monoidal category (so in particular all $R$-module categories since they're strict), but I am hopeless with all of that. Also, I don't understand the issues surrounding $E_\infty$-structures; it seems like this sometimes depends in what category of spectra you're working in.

3) In Weibel's K-book pg 79, a non-connective version of the K-theory spectrum due to Bass is given, basically by homotopizing the fundamental exact sequence. Do the structures (ring, $E_{\infty}$) from the previous questions extend here? (Also, as a digression, this construction seemingly only applies for rings - is there a version of this construct whose input is more categorical?)

4) (a bit more farfetched) For the kinds of rings you find in arithmetic geometry (esp. rings of integers in global fields), is there any relationship between the Bass non-connective spectrum and $K(1)$-localization? cf. the Lichtenbaum-Quillen conjectures implying that such rings are essentially $K(1)$-local.

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Have you looked at Chapter VI of EKMM? It's all about the algebraic K-theory spectrum. The intro says it was part of Mandell's PhD thesis. Indeed, Theorem 6.1 in that chapter seems to answer your (1) and (2). It says the algebraic K-theory spectrum KR is equivalent to an $E_\infty$ ring spectrum.

Also, $E_\infty$ does not depend on the model you use for spectra. You probably know Schwede's result that the stable homotopy category is rigid. It's also monoidally rigid (I think this was proven by Schwede and Shipley). Furthermore, because $E_\infty$ is encoded by a $\Sigma$-cofibrant operad, any monoidal Quillen equivalence lifts to a Quillen equivalence of $E_\infty$-algebras. See the Schwede-Shipley paper on monoidal equivalences.

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  • $\begingroup$ Ah, thank you! This is just what I need. I'll accept this answer; maybe I should split up the other two parts into another question. $\endgroup$ – xir May 2 '18 at 1:00
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    $\begingroup$ You should also look at later papers by Elmendorff and Mandell. $\endgroup$ – Neil Strickland May 5 '18 at 9:22
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Proposition 5.9 in [Blumberg, Gepner, Tabuada. Uniqueness of the multiplicative cyclotomic trace. arXiv:1103.3923] shows that both connective and nonconnective K-theory define lax symmetric monoidal functors (on the infinity-category of stable idempotent-complete infinity-categories). It follows that if you apply either variant of K-theory to a symmetric monoidal stable idempotent-complete infinity-category, you end up with an $E_\infty$-ring spectrum. Of course this applies to the infinity-category of perfect complexes on a commutative ring or $E_\infty$-ring spectrum.

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  • $\begingroup$ Thanks so much! If only I could accept two answers. $\endgroup$ – xir May 8 '18 at 3:02

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