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Let us try to figure out what's happening on discrete rings, where $E_2=E_\infty$. The category $\mathrm{LMod}^{(2)}$ is, as you surmised, the category of pairs $(A,B)$ where $A$ is a commutative alge …

The answer is yes, and this follows essentially from the Jordan decomposition of nilpotent endomorphisms. Let $(F^n,\nu)$ be an $n$-dimensional vector space and a nilpotent endomorphism. Then $\nu^n=0 …

No, the definition in Schlichting's first paper are not the "correct" definition of higher Witt groups (in any case they are not the analogue of Balmer's Witt groups), rather they are some shifted hig …

Fibrant replacement is essentially sheavification with respect to the corresponding topology. So étale K-theory is nothing more than the part of K-theory that satisfies étale descent. Concretely (and …

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least …

I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks …

I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the …

As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.

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_\i …