# About embedding pure motives into the triangulated category of mixed motives and some further questions about motivic cohomology

I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. I don't know if I can post a lot of questions in the same post and if they are suitable for the site, if not, just delete it.

What fails if I try the same construction for non-smooth stuff, i.e., if I consider $DM^{eff}_{Nis} (k) =D^{-}Sh_{Nis} (Corr_k)[W^{-1}$] where $W$ contain all the cones $\mathbf{Z}_{tr} (X \times \mathbb{A}^1) \rightarrow \mathbf{Z}_{tr} (X)$ for $Ob (Corr_k) =Ob( Sch_k$)?

What is $\mathbb{L}$ sitting inside the triangulated category of mixed motives? (I think it's simply $M(1) = \mathbf{Z}(1)= C_* \mathbf{Z}_{tr} (\mathbb{G}_m) [-1]$, but I'm not sure). How is this embedding explicitly (What's the functor $Mot^{eff}_{\sim}(k) [\mathbb{L}^{-1}] \hookrightarrow DM(k)$ for $\sim$ some relation on cycles) ? (I have an idea of how to construct it, but I have never seen it written so I'm a little insecure…).

Is there any geometrical meaning for $M(-1)$ and $\mathbb{L}^{-1}$? Furthermore, is this related to the stabilization of spectra?

Is it possible to extend the construction of the category of mixed motives (by Voevodsky) over general (non Noertherian) base schemes? If not in general, are there any explicit conditions?

Is the smash product of pointed schemes geometric (in some specific cases) in the sense that there exists some scheme such that $M(Y) = M((X_1, x_1) \wedge… \wedge (X_n, x_n))$ ? (Maybe resembling what would be a motive for some scheme $Y$ not smooth... ). If not, how should I think about the smash product (what picture should I have in mind?)?

What's the intuitive (geometrical) meaning of shifting back in the definition $\mathbf{Z}(q) = C_* \mathbf{Z}_{tr} (\bigwedge^q\mathbb{G}_m) [-q]$? Or, more generally, what's the intuition (in some geometric sense) of $\mathbf{Z}(q)$? (Maybe using the Eilenberg-Maclane spectrum gives a more geometrical interpretation of $\mathbf{Z}(q)$…)

• This post requires some clarification. For instance, what do you mean by "try the same construction" in the first question. The "How is this embedding?" in the second complex of questions is also unclear, you are talking about an object in a category. The answers to your questions will likely depend on coefficients - are you interested in integral, rational, finite coefficients? Commented Oct 5, 2014 at 9:43
• @MatthiasWendt By the same construction I mean the one using the derived category of complexes of sheaves over smooth schemes with a fixed base with the Nisnevich topology (or the simplicial sheaves in the nisnevich site of smooth schemes). About the embedding I'm asking explicitly how is the functor (on abjects it's clearly $M(X)$, but for morphisms…). And I'm interested mainly in integral coefficients, but if anyone have further ideas about the other coefficients it would be useful too. Hope i have clarified it. Commented Oct 5, 2014 at 11:30
• My impression is that your question is quite sensitive to the choice of notation, whereas the notation widely varies from paper to paper. So I would recommend you to write explicitly which texts you are trying to read. Commented Oct 5, 2014 at 12:54
• By the way, @PeterArndt, sorry for bothering you, but any idea? Commented Oct 7, 2014 at 22:08

Ok, let me see if I can shed light on some of the questions raised.

"What fails if I try the construction with non-smooth stuff?" This question is a bit unspecific, a lot of things can fail, depending on what you want. For instance, if you want higher Chow groups to come out of the construction, you need homotopy invariance which is false for non-smooth schemes.

For the construction of categories of motives starting from a site of (possibly non-smooth) schemes, you would usually use some topology like $h$-, $qfh$- or $cdh$-topology. In these topologies, schemes will be locally smooth because some sort of resolution of singularities gives you coverings. In some cases (best with rational coefficients), there are comparison results to the Nisnevich topology.

The Lefschetz motive questions: If you take Voevodsky's embedding from Chow motives into $DM_{gm}$, the image of the Lefschetz motive is the Tate motive $\mathbb{Z}(1)[2]$. This is the reduced motive of $\mathbb{G}_m$ or $\mathbb{P}^1$, appropriately shifted, i.e., either $\ker\left(C_\ast\mathbb{Z}_{tr}(\mathbb{G}_{m,S})\to C_\ast\mathbb{Z}_{tr}(S)\right)$.

The embedding from Chow motives to Voevodsky motives is given by mapping a variety $X$ to $M(X)=C_\ast\mathbb{Z}_{tr}(X)$. Since Voevodsky's category is idempotent complete and has transfers, this assignment factors through Chow motives. Hence, the embedding takes a Chow motive $(X,p)$ to the summand of $M(X)$ given by the projector $p$. I should think that this is discussed in the red book (Friedlander, Suslin, Voevodsky: Cycles, transfers and motivic homology theories), at least the book contains the theorem that Chow motives embed as full subcategory of Voevodsky motives.

I would say that there is no geometric meaning to $\mathbb{Z}(-1)$ (unless you consider virtual bundles as something geometric). It is simply constructed as the $\otimes$-inverse of $\mathbb{Z}(1)$, in the same way as spectra arise from CW-complexes by inverting $S^1$ w.r.t. the smash product. Somehow, all this technical stabilization business is necessary because there is no geometric meaning to $\mathbb{Z}(-1)$ (although this is more like a philosophical belief).

The construction of motives over more general bases is currently being studied. Check out the works of Joseph Ayoub, cf. his homepage where you can find his papers. His ICM paper gives a nice overview of the state of the art. I guess the easiest construction of motives over arbitrary bases is via étale sheaves made $\mathbb{A}^1$-invariant and $\mathbb{P}^1$-stable. For the resulting category, a full six functor formalism is available...

There are also constructions using Nisnevich sheaves with transfers, cf. the work of Cisinski-Déglise, but the construction of these categories is a bit complicated in full generality because of difficulties with the intersection theory. With rational coefficients, these categories can be compared to the étale version; with integral coefficients, there are problems with the localization triangle.

Representability of smash products: I do not know of any results, but I would guess that the smash product is more likely not represented by a scheme. I am more confident saying that the smash product is most likely not represented by a smooth scheme: there is a conjecture of Asok stating that the spheres $S^{p,q}\cong S^p\wedge\mathbb{G}_m^{\wedge q}$ are not $\mathbb{A}^1$-homotopic to smooth schemes unless $p=2q$ or $p=2q-1$. Admittedly, that's a bit different from the corresponding question about motives... The right way to think about the smash product is: it's the same thing as in topology, same categorical properties etc.

Finally, I would again say that there is no geometric meaning to shifting $\mathbb{Z}(q)$. The topological analogues are exactly the Eilenberg-Mac Lane spaces, one should think of $\mathbb{Z}(q)[n]$ as the motive of $K(\mathbb{Z}(q),n)$. In topology as well, there is no real geometric relation between $K(\mathbb{Z},n)$ and $K(\mathbb{Z},n-1)$...

I would also like to point out that $\mathbb{Z}_{tr}\left(\bigwedge^n\mathbb{G}_m\right)$ does not really have a meaning, because $\mathbb{Z}_{tr}$ takes a scheme and turns it into a sheaf with transfers - you should take $\mathbb{Z}(1)$ and then take its smash powers, not the other way round.

• I would rather specify that $L$ becomes $Z(1)[2]$ in Voevodksy's notation. Besides, it seems that Cisinski-Deglise's are now able to describe (relative) motives over characteristic $0$ varieties using the cdh-topology. Commented Oct 5, 2014 at 12:49
• Oh, yes, fixed that. You can not shift in Chow motives, so you can only take the reduced motive of $\mathbb{P}^1$, which is $\mathbb{Z}(1)[2]$. Commented Oct 5, 2014 at 12:51
• Also, my impression that the paper math.univ-toulouse.fr/~dcisinsk/DMet.pdf (by Cisinski and Deglise; unfortunately the link does not work at the moment) currently gives more information on \'etale motives that Ayoub's texts. Commented Oct 5, 2014 at 12:58
• Thanks for the answer. I tried editing again. Hope now you can understand it. Commented Oct 7, 2014 at 22:06
• @user40276: yes, it's better now. I added a discussion of the embedding of Chow motives into Voevodsky motives. For the other questions, I said most of what I can in a short post. As there are recurrent questions on geometric interpretation: most of the constructions like smash products etc are not geometric, these are constructions not available for schemes. They are added to allow for arguments of a more topological flavour. Commented Oct 8, 2014 at 19:26