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My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result wa …

There may be some confusion in this question about what exactly Voevodsky/Ayoub and Mann do, as they do very different things. Mann's thesis constructs a formalism of six operations in the setting of …

It is self-dual. In general the dual of a smooth proper $R$-linear $\infty$-category $C$ is always $C^{\operatorname{op}}$, but for a scheme $X$ we have $\operatorname{Perf}(X) = \operatorname{Perf}(X …

Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an $\in …

This holds if the characteristic of $k$ is not 2, and it follows from the Milnor conjecture proved by Voevodsky. Voevodsky ultimately proved the following (Theorem 6.17 in https://annals.math.princeto …

Your description of the six functors does not mention any relations between the $!$-functors and the $*$-functors or the tensor product, which is where these dualities are hiding. Poincaré duality is …

Such an "étale universal cover" exists at least if $X$ is Noetherian and geometrically unibranch (and for all qcqs $X$ if one considers profinite étale homotopy types). Background. I will regard the é …

I have seen the morphishm $\nu\colon (A^{\otimes n})^{\Sigma_n}\to k$ in a couple of places: On page 81 of A. Suslin, V. Voevodsky, Singular homology of abstract algebraic varieties It is defined wh …

The formula also holds for perfect complexes. This can be deduced from the case of vector bundles, although it requires a lot of structure in that case. Namely, we need to use the fact that the determ …

It is certainly true that descent implies hyperdescent whenever $\mathcal C$ is a $n$-category for some $n<\infty$ (it wasn't clear from your question whether you knew this or not). This is because, f …

Yes, the two $\infty$-topoi are equivalent. Let $u: \mathrm{Aff} \to \mathrm{AlgSp}$ be the inclusion. Then $u$ preserves étale covering families and pullbacks, hence commutes with the formation of th …

This is not true even if $\mathcal X$ is an Artin stack. For example, let $G$ be a smooth group scheme over the base $T$, and let $\mathbf BG$ be its classifying stack (the category of $G$-torsors fib …

The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player …

Here's an easy way to see this (which is more or less an elaboration of Mikhail's answer): Suppose $i > 0$ and $n\in \mathbb Z$. We have the Thom isomorphism $$ H^n(X, \mathbb Z(-i)) \cong H^{n+2i}_X …

I'm not sure if that's exactly what you mean by "being motivic", but this long exact sequence comes from a triangle in Voevodsky's category of (integral) motives $DM(\mathbb Q)$. More generally if $ …