Why is the motivic category defined over the site of smooth schemes only?

Question

Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?

As a casual observer, I see two broad classes of possible reasons:

• Technical reasons: Perhaps the Nisnevich topology doesn't make sense over non-smooth schemes, or maybe $\mathbb{A}^1$ or $\mathbb{G}_m$-locality become more complicated over non-smooth schemes.

• Goals of the construction: Perhaps the stable and unstable categories make perfect sense over non-smooth schemes, but maybe motivic cohomology or algebraic K-theory simply don't extend to stacks over the resulting category.

In either case, I wonder: is it clearly impossible, by some modification of the construction of these categories, to get things to work over non-smooth schemes? Would this be desireable?

Answer 1

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 Nisnevich topology you need étale extensions of affine spaces in your category, and every smooth $S$-scheme is (Zariski) locally such.

That said, I can think of two fundamental places in the theory where smoothness is crucial, both of which also showcase the relevance of the Nisnevich topology and of $\mathbb A^1$-homotopy.

(1) The first is the localization property already addressed in Adeel's answer, which is itself crucial for many things, such as the proper base change theorem. A characterizing property of henselian local schemes $S$ (which are the points of the Nisnevich topology) is that for $f:X\to S$ étale every section of $f$ over the closed point $S_0\subset S$ lifts uniquely to a section of $f$ over $S$ ("Hensel's lemma"); this is what makes localization work for sheaves on the small Nisnevich or étale sites. If $f: X \to S$ is smooth, it is still the case that every section $s_0:S_0 \hookrightarrow X$ lifts to $S$ (this uses smoothness in an essential way), but not uniquely. However, once a lift $s:S\hookrightarrow X$ has been chosen, then $X$ can be presented as an étale neighborhood of $S$ in the normal bundle of $s$, and it follows that the Nisnevich sheafification of the "space of lifts" of $s_0$ to $S$ is $\mathbb A^1$-contractible. Thus, in some precise sense, lifts are still unique up to $\mathbb A^1$-homotopy. This is the proof of the Morel-Voevodsky localization theorem in a nutshell.

Another nice consequence of the localization property is that fields form a "conservative family of points" in motivic homotopy theory (at least for the $S^1$-stable theory, though one can also say something unstably), something that could not be achieved using a Grothendieck topology alone.

(2) The second is Cousin/Gersten complexes. This is now specific to the case of a base field $k$, in which localization is useless. The key input here is a geometric presentation lemma of Gabber, a statement of which can be found as Lemma 15 in the introduction to Morel's book (http://www.mathematik.uni-muenchen.de/~morel/Prepublications/A1TopologyLNM.pdf). A consequence of this lemma is that every $\mathbb A^1$-invariant Nisnevich sheaf $F$ (of spaces or spectra, say) is "effaceable" on smooth $k$-schemes, in the sense of Colliot-Thélène-Hoobler-Kahn (https://webusers.imj-prg.fr/~bruno.kahn/preprints/bo.dvi). This implies that the coniveau spectral sequence degenerates at $E_2$ and hence that the Cousin complex $$ 0 \to \pi_nF(X) \to \bigoplus_{x\in X^{(0)}} \pi_{n}F_x(X_x) \to \bigoplus_{x\in X^{(1)}} \pi_{n-1}F_x(X_x) \to \dots $$ is exact when $X$ is smooth local. Here, $X^{(n)}$ is the set of points of codimension $n$, $X_x=\operatorname{Spec}(\mathcal O_{X,x})$, and $F_x(X_x)$ is the homotopy fiber of the restriction map $F(X_x) \to F(X_x-x)$.

These Cousin complexes are the basis for many computations in motivic homotopy theory, for example in the above-mentioned work of Morel. They can perhaps be viewed as a replacement for cell decompositions in topology.

So, in summary, the smooth/Nisnevich/$\mathbb A^1$ combo simultaneously allows us to (1) reduce questions over general base schemes to the case of base fields via localization, and (2) perform interesting computations in the latter case.

Answer 2

It makes sense to consider larger versions of the (unstable and stable) motivic homotopy categories built out of the site $Sch_S$ of all schemes over $S$ (say of finite type to avoid dealing with size issues). Denote the stable variant by $\underline{SH}(S)$; there is a fully faithful functor $SH(S) \hookrightarrow \underline{SH}(S)$ which identifies the usual motivic homotopy category as a right localization of the larger variant.

One of the main pleasant features of the categories $SH(S)$ is that they satisfy the localization property (a theorem of Morel and Voevodsky) which means that for any closed immersion $i : Z \to S$ with open complement $j : U \to S$, the diagram $$ SH(Z) \xrightarrow{i_*} SH(X) \xrightarrow{j^*} SH(U) $$ is an "exact sequence" in the sense that $i_*$ is fully faithful and its essential image coincides with kernel of $j^*$. This property gives rise to the usual long exact localization sequences in motivic cohomology, algebraic cobordism, and so on. Further, the localization property is one of the main inputs into the existence of the formalism of Grothendieck's six operations on the categories $SH(S)$ (constructed by Voevodsky and Ayoub). For example, part of the formalism of six operations are the proper base change and projection formulas, and the localization property demonstrates these in the case of closed immersions.

However, the larger categories $\underline{SH}(S)$ fail to satisfy the localization property. The problem is that when you work with the site $Sch_S$, you "automatically" get a functor $i_\sharp$, left adjoint to $i^*$, which satisfies base change against the operation $f^*$. By adjunction this means that $i^*$ satisfies base change against $f_*$. Using this, it is easy to see that the localization property cannot hold for $\underline{SH}(S)$ (see Remark 2.4.4 here). This failure can be interpreted as saying that the functor $j_*$ on $\underline{SH}(S)$ is "wrong" (that is, it does not agree with the usual $j_*$ under the embeddings $SH(S) \hookrightarrow \underline{SH}(S)$).

One reason the construction $\underline{SH}(S)$ is interesting is that it can be used to define a cdh-local version of $SH$ (which cannot be done using the smooth site $Sm_S$, since you need non-smooth schemes to make sense of the cdh topology). Namely, let $\underline{SH}_{cdh}(S)$ denote the version of $\underline{SH}(S)$ formed with the cdh-topology instead of the Nisnevich topology; then let $SH_{cdh}(S)$ denote the full subcategory generated under colimits by (cdh-localizations of) smooth $S$-schemes. One can then prove that the canonical functor $SH(S) \to SH_{cdh}(S)$ is an equivalence ("motivic spectra have cdh-descent").