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You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles. You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) schem …

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 …

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 …

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 …

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 …

There's a free-forgetful adjunction between $SH_s(k)$ and $DM^{eff}(k)$, where $SH_s(k)$ is the category of $S^1$-spectra (as opposed to $\mathbb{P}^1$-spectra). The right adjoint simply takes a sheaf …

Voevodsky's statement (1) is plainly false if $k$ is not of finite transcendence degree over $k_0$. For then the fraction field of any $O_{X,x}$ is of infinite transcendence degree over $k_0$ whereas …

I believe the answer is YES and, more generally, that $\tau_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$- …

The only reference I know for Künneth theorems in this generality are SGA4 and SGA4.5. Specifically, SGA4 has theorems for étale cohomology with proper support (which hold in ridiculous generality), b …

Yes, you can reduce to the finite type case by noetherian approximation (Appendix C in Thomason-Trobaugh). Namely, you can write $X=lim_\alpha X_\alpha$ where $X_\alpha$ is of finite type over $k$. Th …

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 …

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 …

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 …

There certainly is a notion of higher Tannakian category which would have meaningful higher homotopy groups. I'm not sure how much of the theory has been worked out already, but higher Tannakian duali …

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 …