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The six functor formalism applies to $D$-modules, but you need to extend the theory to possibly non-smooth schemes. For this, we see that hypersheaves on the site of pairs $(X,Z)$, with $Z$ a closed s …

To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $ …

There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of fin …

There is an homotopy theory associated to any geometry. This can be done in various ways. but a rather systematic point of view is the one of Morel and Voevodsky. The idea is that a geometry should de …

If you have an abelian category $A$ with enough injective, for any functor defined on bounded below cochain complexes $\Phi\colon C^+(A)\to D$ sending cohain homotopy equivalences between degree-wise …

The triangulated category $\mathsf{D}_{\mathsf{B}}(\mathsf{A})$ can be promoted to a stable $\infty$-category. One of the many interests of working with stable $\infty$-categories is that we have a re …

There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is …

Here is a long comment about the étale setting. If $X\to Y$ is a universal homeomorphism of schemes (or Deligne-Mumford stacks), then it induces an equivalence of small étale sites, and thus of topoi: …

The only condition you need on $X$ for such a formula to hold is that it is coherent (=quasi-compact and quasi-separated). Let $R$ be a discrete ${\mathbf{Z}_\ell}$-module. We consider the sheaf $\und …

If you work in equal characteristic, you have a chance of survival. In characteristic zero, this amounts to check that Chow groups do see nilpotent extensions. In characteristic $p>0$, if $p$ is inver …

A precise way to say that "sufficiently nice functors" from schemes to complexes of vector spaces determine realization functors of Voevodsky's motives is the notion of mixed Weil cohomology. There a …

The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the …

Let $E(S)$ be the category of Nisnevich sheaves on the site of smooth schemes over some base $S$. Then Morel and Voevosy's homotopy category $\mathrm{H}(S)$ is obtained as a localization of the catego …

Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of Lojasiewicz, this is true for any semi-algebraic set (one ca …

I assume the question holds in contexts where we can glue open immersions and proper morphisms to produce $f_!$ for $f$ separated of finite type. In particular, we shall have $f_!=f_\ast$ for $f$ prop …