Voevodsky's Triangulated Categories of Motives and their Relationships

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint.

Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(!; see below) with rational coefficients "do not depend on the choice of the topology" (whether one takes Zar, Nis, Et, cdh, qfh, or h). Next, it is easier to prove "nice properties" of motives if one takes "the strongest possible" topology. Since one expects the etale realization of motives to be conservative, and etale cohomology satisfies h-descent, it is reasonable to consider h here.

As about $DM$ being isomorphic to $D^b(MM)$: one starts with the question of the existence of the motivic t-structure for DM. Its existence depends on the known relation of DM to the so-called motivic cohomology (and so, also to Chow motives) and on several hard conjectures; see the papers: Beilinson A., "Remarks on Grothendieck's standard conjectures"; Hanamura M., "Mixed motives and algebraic cycles, III".

To proceed further towards isomorphism to $D(MM)$ one needs the so-called $K(\pi,1)$-property for the motivic $t$-structure; I don't know any papers on this subject (yet you can look for them).

Now let's consider motives with integral coefficients. If one wants the morphisms in DM to calculate the so-called motivic cohomology (= higher Chow groups) then et/qfh/h topologies should be abandoned. One can take Nis or even Zar instead; yet instead of "arbitrary" sheaves of abelian groups one should consider the so-called sheaves with transfers. Nis is a more convenient choice (and the cdh-topology is an important tool also).

I won't embark on the difficult question of what one wants out of a category of motives, but I can make some comments on what might motivate the various choices of topologies.

Nisnevich (aka completely decomposed, aka completely decomposed étale). If one defines cohomology with support of a closed subspace $Z \subseteq X$ in such a way to induce long exact sequences $\dots \to H^n(X) \to H^n(X{-}Z) \to H^{n-1}_Z(X) \to H^{n-1}(X) \to \dots$, then the Nisnevich topology (on the category of smooth schemes) makes sure that for any regular closed immersion of smooth $k$-varieties $Z \to X$, there is an isomorphism $$ H_Z^\bullet(X) \cong H_{\mathbb{A}^{d-c}}^\bullet(\mathbb{A}^d) $$ when $X$ and $Z$ are of pure dimension $d$ and $d-c$ respectively.

(Notice that this statement needs smooth schemes to work).

One example of a very serious consequence of this is the Gersten exact sequence for homotopy invariant sheaves with transfers [Voevodsky, Coh.th.of.presh.w.transfers, Theorem 4.37]. Others include Localisation [Morel-Voevodsky, $\mathbb{A}^1$-hom.th.of.sch., Theorem 3.2.21], and Purity [Morel-Voevodsky, $\mathbb{A}^1$-hom.th.of.sch., Theorem 3.2.23].

For other benefits of the Nisnevich topology (cohomological dimension $=$ Krull dimension, $K$-theory has descent, direct image is exact) see [Morel-Voevodsky, $\mathbb{A}^1$-hom.th.of.sch., Beginning of Section 3].

Finite (a.k.a.fs, a.k.a. finite surjective). This is the topology on the category $Sch/k$ of separated finite type schemes whose covering families are jointly surjective families of finite morphisms. It is a way to build in transfers, in the sense that the canonical functors $$Shv_{fs}(Cor/k, \mathbb{Z}) \stackrel{\sim}{\to} Shv_{fs}(Sch / k, \mathbb{Z})$$ $$PreShv(Cor/k, \mathbb{Q}) \stackrel{\sim}{\leftarrow} Shv_{fs}(Sch / k, \mathbb{Q})$$ are equivalences of categories. (The situation is more subtle if one wants to use only smooth schemes).

One important use of this is to compare "cycle theoretic" theories (like Suslin homology, or Bloch's higher Chow groups) with "sheaf theoretic" theories (like étale cohomology). Cf. [Suslin-Voevodsky, Sin.hom.of.ab.alg.var.] where they prove that algebraic singular homology $H^{sing}_i$ (now called Suslin homology $H^S_i$) can calculate the classical topological singular homology (which can be calculated via sheaf cohomology) for $\mathbb{C}$-varieties $X$.

$$ H_i^{S}(X, \mathbb{Z} / n) \cong H_i(X(\mathbb{C}), \mathbb{Z} / n) $$

Cf. also Suslin's article "Higher Chow groups and étale cohomology" where he shows that Bloch's higher Chow groups calculate étale cohomology with compact support for equidimensional quasi-projective varieties over algebraically closed fields $k$ with char.$k \nmid m$, $i \geq d = $dim $X$.

$$CH^i(X, n; \mathbb{Z} / m) \cong H_c^{2(d-i) + n}(X, \mathbb{Z}/m(d-i))^\# $$

The qfh-topology. This is the topology generated by the Zariski topology, and the finite topology. $$ \mathrm{qfh} = \langle \textrm{finite surjective, Zariski} \rangle. $$ Remark: It is finer than the étale topology, so we have qfh $= \langle $fs, Zar$ \rangle = \langle $fs, Nis$ \rangle = \langle $fs, étale$ \rangle$.

Proper cdh (which should be called the cdp $=$ completely decomposed proper topology). Every scheme admits a Zariski covering by affine schemes. So properties of Zariski sheaves are completely determined by their values on affine schemes. Analogously, (if the base field admits resolution of singularities) every (finite type) $k$-scheme admits a proper cdh covering by smooth $k$-schemes. So just as we have $$ Shv_{Zar}(Aff/k) = Shv_{Zar}(Sch /k) $$ we have (if res.of.sing. holds for $k$) $$ Shv_{cdp}(Sm/k) = Shv_{cdp}(Sch /k). $$ (the sites on the left are equipped with the induced topologies).

Cohomologically, (and in general, i.e., without any dependance on res.of.sing) for any proper birational morphism (i.e., blowup) $X' \to X$ which is an isomorphism outside of a closed subscheme $Z \subset X$, setting $Z' = Z \times_X X'$ there is canonical long exact sequence $$ \dots \to H^n_{cdp}(X, F) \to H^n_{cdp}(X', F) \oplus H^n_{cdp}(Z, F) \to H^n_{cdp}(Z', F) \to H^{n + 1}_{cdp}(X, F) \to \dots $$ for any cdp sheaf $F$, and the same is true of any topology finer than the cdp topology (e.g., cdh, h, see below). In fact, the cdp topology is the coarsest topology with this property. In fact, if a strong version of resolution of singularities holds, the cdp topology is the coarsest topology with the above long exact sequence for blowups of smooth schemes in smooth centres.

cdh (a.k.a., completely decomposed h-topology). The cdh topology is generated by the proper cdh topology and the Nisnevich topology. $$ \textrm{cdh} = \langle \textrm{Nisnevich}, \textrm{proper cdh} \rangle. $$ So we get the above cohomology with supports property for smooth closed immersions, the blowup long exact sequence, and, assuming res.of.sing. the power to reduce arguments to smooth schemes.

Inconsequential remark: The name is a slight misnomer. We have $h = \langle$ étale, proper $\rangle$, and $cdh = \langle$ comp.dec.étale, comp.dec.proper $\rangle$, but while every cdh covering is a completely decomposed $h$-covering, the converse is not true, comp.dec.h $\neq$ cdh. There are completely decomposed flat coverings which are not cdh coverings.

Proper. The topology on $Sch/k$ whose coverings are jointly surjective families of proper morphisms. Without assuming resolution of singularities, a theorem of de Jong on alterations implies that every $k$-scheme admits a proper covering by smooth schemes. $$ Shv_{\textrm{proper}}(Sm/k) = Shv_{\textrm{proper}}(Sch /k).$$

The h-topology. The $h$-topology is generated by the proper topology and the Zariski topology. $$h = \langle \textrm{proper}, \textrm{Zariski} \rangle.$$ It is generated by a number of other combinations of the previously mentioned topologies, e.g., $h = \langle$cdh, finite$\rangle = \langle $proper, étale $\rangle$.

$L$dh (a.k.a. $L$-decomposed topology) The proper topology has the disadvantage from the cdh topology that many interesting cohomology theories satisfying cdp descent, only satisfy proper descent after passing to rational coefficients. Between cdp and proper lies the $L$dp-topology. If $L$ is a collection of primes, then one defines a morphism $f: Y \to X$ to be $L$-decomposed if for every (not necessarily closed) point $x \in X$, there exists a point $y \in f^{-1}\{x\}$ such that $k(y) / k(x)$ is a finite field extension of degree prime to every element of $L$. If $L = \{\ell\}$ then we just write $\ell$dp. We have $\varnothing$dp $=$ proper, and $\mathbb{P}dp = cdp$ if $\mathbb{P}$ is the set of all primes.

It follows from a theorem of Gabber that every for any choice of $\ell \neq $char.$k$, every (finite type) $k$-scheme admits an $\ell$dp-covering by smooth $k$-schemes. Many cohomology theories of interest satisfy $\ell$dp descent after passing to $\mathbb{Z}_{(\ell)}$-coefficients, so we can still obtain information about their $\ell$-torsion using this topology.

Just as we have $cdh = \langle $Nis, cdp$\rangle$ and h $= \langle$ Nis, proper $\rangle$, we could define $$ \ell dh = \langle \textrm{Nis}, \ell dp \rangle. $$ However, in practice, it is much more useful to use the definition $\ell$dh $ = \langle$ cdh, fps$\ell' \rangle$ where fps$\ell'$ is the topology whose coverings are $\ell$-decomposed families of finite flat surjective morphisms, see below. This way questions about $\ell$dh descent be be broken up into a cdh descent part and an fps$\ell'$ descent part.

Here is the relationship: $$ \begin{array}{cccc} \textrm{Set of primes} & \textrm{Topology} & \textrm{Coefficients} & \textrm{Theorem} \\ \hline \textrm{all primes} & cdh & \mathbb{Z} & \textrm{Hironika's res.of.sing.} \\ \{\ell\} & \ell dh & \mathbb{Z}_{(\ell)} & \textrm{ Gabber's theorem on alterations } \\ \varnothing & h & \mathbb{Q} & \textrm{ de Jong's theorem on alterations } & \end{array} $$

The fps$\ell'$ topology. This is a version of the finite topology. It is generated by morphisms which are finite flat surjective of degree prime to $\ell$, where $\ell$ is a prechosen prime number. The point is that it can be much easier to show that a cohomology theory has "trace" morphisms for finite flat morphisms, than that it has transfers. If one defines an appropriate notion of presheaves with traces (cf. https://arxiv.org/abs/1305.5349) then it is easy to show that every presheaf of $\mathbb{Z}_{(\ell)}$-modules with traces is an fps$\ell'$ sheaf, and one can leverage this to show an equivalence of categories $$ Shv_{cdh}(Cor/k, \mathbb{Z}_{(\ell)}) = Shv_{\ell dh}(Cor/k, \mathbb{Z}_{(\ell)}) $$ Note, the same is true with h and $\mathbb{Q}$ replacing $\ell$dh and $\mathbb{Z}_{(\ell)}$.

Some comments regarding how these categories are related:

In reasonable situations, all your categories should satisfy the gluing property for an open subscheme and closed complement. This property plus zariski descent implies cdh descent, whereas gluing plus etale descent implies h descent. (In fact as I recall h descent is equivalent to etale descent plus invariance under passing to the reduced closed subscheme structure - not quite sure but I think something like this is true.)

Also etale descent is more or less the same as nisnevich plus Galois descent. Transfers plus $n$ invertible implies degree n Galois descent (more or less).

This explains how the nisnevich version embeds into the big cdh version (big meaning built from all schemes, not just smooth ones), and similarly etale version embeds into big h. These embeddings become equivalences if your base has resolution of singularities with respect to the coefficients (in some sense), so e.g. over a field of characteristic zero (hironaka) or over a perfect field of characteristic p after inverting p (De Jong). This also explains why with rational coefficients all the categories agree.

In the presence of transfers, zariski descent plus homotopy invariance implies nisnevich descent. But this is not true without transfers, because then you do not get gluing (which is more or less equivalent to homotopy purity). However if you put in gluing (or homotopy purity) by hand then you should get an equivalent category.

Finally you could consider motives without transfers. Here with rational coefficients you again get the same category, at least over a field of finite etale dimension. In the infinite dimensional case the rational category splits into plus and minus part and plus is DM. Etale locally the minus part vanishes but in the nisnevich topology this is not the case. In the etale topology (or finer) you also get the same category integrally. This reduces to a rational and a torsion problem. Rationally there is not a problem (as I just said) and for torsion coefficients we have rigidity (I.e. everything collapses and you just get ordinary etale cohomology on the small site).

Motives without transfers in the nisnevich topology with torsion coefficients are essentially mysterious and almost certainly not the same as with transfers, even over a field with finite etale dimension.

I'm currently on mobile so I cannot give references.