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(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understan …

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've decided to expand my comment into an answer. The point is what do you think a topology is for. If you think that a topology is for talking about convergence of sequences, then no the Zariski top …

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 …

Your first map fits in an action of the Steenrod algebra. In fact $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$, an element of the derived category …

TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as …

Let's start approaching the question from the simplest possible case $Y=*$. What should be the points of $[X/G]$? Recall that the idea here is to generalize the construction of the action groupoid fo …

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 …

One way of finding a "fully derived" version of Poincaré duality is Atiyah duality. This says that for any closed manifold $M$ there is an equivalence of spectra (in the sense of algebraic topology) $ …

What is true is that there is an antiequivalence between the category of schemes affine over $S$ (that is $S$-schemes for which the preimage of an open affine of $S$ is an open affine) and quasi-coher …

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 …

While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated? The answer is, of course, that there's no …

The étale topos of a field $k$ is just the topos of sets with a continuous $\mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-cat …

The derived category of perfect complexes is idempotent complete, because it is the sub category of compact objects in the derived category of quasi coherent sheaves (which is idempotent complete by t …

This question is strongly related to this question. However it seems to me sufficiently distinct to warrant asking it separately. Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞- …