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The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck. Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly …

I assume that $Z$ is also smooth over $k$. Then the base change of the map $$Bl_Z(X)-\sigma(Z)\to X$$ to $Z$ is the map $$\mathbb{P}(N_{X,Z})-\sigma(Z) \to Z$$ where $\mathbb{P}(N_{X,Z})$ is the …

The answer to both question is yes, provided you invert $p$ in characteristic $p$ (though conjecturally this is not necessary). In fact, as far as I can see, all of Spitzweck's computations apply to f …

Here's a direct link to the book by Hovey–Palmieri–Strickland. The category of motivic spectra is known to satisfy the axioms of Definition 1.1.4 in the book when the base is a countable field of cha …

Singular cohomology is represented by an $E_\infty$ motivic ring spectrum. That spectrum is $\mathbf{R}f(H\mathbb{Z})$ where $f$ is right adjoint to the stable topological realization functor and $H\m …

Yes. A bit more generally, if $\xi$ is a perfect complex of rank $\geq 0$, then $Th(\xi)$ is effective (even very effective): the question is Nisnevich-local on $X$ and $\xi$ is locally a complex of …

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 …

It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the N …

(1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that $$MGL^{p,q}(X)=0$$ if $p>q+dim …

$MGL$ does not admit a structure of $H\mathbb Z$-module. There are many ways to prove this. As Sean said in the comments, if it were true over $\mathbb C$, topological realization would imply that $MU …

My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF.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