Homotopy theory of schemes examples

Question

Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?

Answer 1

To keep things simple, let us assume we work over a perfect field. The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group $\mathbb G_m=\mathbb A^1-\{0\}$ has the $\mathbb A^1$-homotopy type of the infinite dimensional projective space. Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as $[X,B\mathbb G_m]=\text{Pic}(X)=CH^1(X)$, where $[?,?]$ stands for the $\text{Hom}$ in the $\mathbb A^1$-homotopy category of $k$-schemes, denoted by $H(k)$.

In general, we denote by $K(\mathbb Z(n),2n)$ the $n$-th motivic Eilenberg-MacLane space, i.e. the object of $H(k)$ which represents the $n$-th Chow group in $H(k)$: for any smooth $k$-scheme $X$, one has

$$[X,\Omega^i K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n))$$

(where $\Omega^i$ stands for the $i$-th loop space functor). For $i=0$, we just get the usual Chow groups:

$$H^{2n}(X, \mathbb Z (n)))\simeq CH^n(X) .$$

Then, there are several models for $K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that $K(\mathbb Z(1),2)$ is the infinite projective space. $K(\mathbb Z(0),0)$ is simply the constant sheaf $\mathbb Z$. For higher $n$, here is a construction (this is Voevodsky's).

Given a $k$-scheme $X$, denote by $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite correspondences from $V$ to $X$ (i.e. the finite linear combinations of cycles $\sum n_iZ_i$ in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in $X$ (I will need this functoriality only for closed immersions).

Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line. The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of $K(\mathbb Z(n),2n)$ in $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i):L(Y)\to L(X)$.

If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of $\mathbb P^1$-spectra, but maybe it is enough at first to think of $\mathbb P^1$-spectra simply as the cohomology theories allowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum $MU$ which represents complex cobordism in algebraic topology). The idea is that $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of $\mathbb P^1$ with coefficients in $E(1)$, is the same as a map of ring spectra $MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories ($MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is $KGL$, the $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if $k$ is of characteristic zero, but is not published yet, and it is known for any field $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).

Answer 2

Motivic cohomology computes Chow groups. And, motivic cohomology is representable in the A^1-category. More specifically, CH^p(X)=H^2p(X,Z(p)). The cohomology groups on the right are representable by "Eilenberg-Mac Lane" spaces in A^1-homotopy theory. Here, by representable, I mean the cohomology groups coincide with homotopy classes of maps to some space.

Some more details: Marc Levine has a paper called The homotopy coniveau filtration; you can find it here. The title refers to a tower that is an analogue of the Gersten resolution in algebraic K-theory. The layers of the homotopy coniveau filtration for the space representing K-theory apparently give the motivic Eilenberg-Mac Lane spectrum.

Let me tell you the details. This is all from the introduction of Levine's paper. Let E be a spectrum (in the stable motivic category). For such a spectrum E, Levine constructs a tower E^(p)->E^(p-1)->...->E. E^(p)(X) is the limit of the spectra with supports E^W(XxA^n) where W is closed of codimension at least p in XxA^n. Then, the layer E^(p/p+1) is the cofiber at level p. When applied to the spectrum representing K-theory, the slice K^(p/p+1)(X) corresponds to Bloch's higher cycle group z^p(X). And, it is well known that this computes (higher) Chow groups and motivic cohomology.

Answer 3

If X is not smooth, then it is possible for the Chow groups and the A^1-represented motivic cohomology theory to disagree.

For instance, if we take X to be two copies of A^1 identified at a point then CH^0(X) has rank 2, but the sheaf represented by X in the A^1 category is contractible, so H^0(X,Z(0)) has rank 1.

To see this last point, we consider X as the colimit of a diagram A^1 <- * -> A^1. Since the maps in this diagrams are monomorphisms of schemes, they are cofibrations. The colimit of the diagram is therefore equivalent to the homotopy colimit, which is invariant under pointwise equivalences of diagrams. Since A^1 is contractible, we are left with the colimit of * <- * -> *, a point.

Morally, A^1 is contractible, so we can shrink down the A^1s without changing the A^1 homotopy type.