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motivic cohomology

Idea

Motivic cohomology was for a long time a hypothetical cohomology theory of schemes whose hypothetical existence Alexander Grothendieck proposed in the 1960s should be the underlying reason for what are called the standard conjectures on algebraic cycles.

Several different realizations of this hypothetical cohomology theory have been proposed:

1. The proposal due to Morel and Voevodsky, based on Morel-Voevodsky’s A1-homotopy theory effectively identifies motivic cohomology with the cohomology as given in the (∞,1)-topos $H_{\mathrm{Nis}}$ of ∞-stacks on the Nisnevich site, following the general notion of cohomology (as described there): for $X$ a scheme and $A\in H_{\mathrm{Nis}}$ some coefficient object, the motivic cohomology of $X$ with coefficients in $A$ is the connected components of the (∞,1)-categorical hom-space of morphisms from $X$ to $A$:

   $$H_{\mathrm{motivic}}(X,A)={\pi }_0{H}_{\mathrm{Nis}}(X,A).$$H_{motivic}(X,A) = \pi_0 \mathbf{H}_{Nis}(X,A) \,.

   More precisely, the full $H_{\mathrm{Nis}}$ knows about what would be called the nonabelian cohomology generalization of motivic cohomology: motivic cohomology proper is the special case of this where coefficient objects $A$ are taken to be spectrum objects with respect to the Tate sphere? built from ${𝔸}^1$ and taken to be A1-homotopy invariant.

   This is described in the section Homotopy stabilization of the (∞,1)-topos on Nis below.

2. Another proposal has been put forward by Voevodsky. More on this in the section Voevodsky’s definition below.

Homotopy stabilization of the $(\mathrm{\infty },1)$-topos on $\mathrm{Nis}$

Write

• $\mathrm{Nis}$ for the Nisnevich site

• $H_{\mathrm{Nis}}$ for the (∞,1)-topos of ∞-stacks on $\mathrm{Nis}$;

• $H_{\mathrm{Nis}}^I$ for its homotopy localization, its A1-homotopy theory;

• ${\mathrm{Stab}}_T(H_{\mathrm{Nis}}^I)$ for the geometric stabilization of $H_{\mathrm{Nis}}^I$ at the Tate sphere?s. This produces the $T$-spectrum objects in $H_{\mathrm{Nis}}$.

Then motivic cohomology, and motivic homotopy are given by connected components of (∞,1)-categorical hom-spaces in ${\mathrm{Stab}}_T(H_{\mathrm{Nis}}^I)$.

There is a standard way to present all this structure:

• as described at models for ∞-stack (∞,1)-toposes, the standard way to present $H_{\mathrm{Nis}}$ is in terms of the model structure on simplicial presheaves on $\mathrm{Nis}$

• then the homotopy localization is modeled by the corresponding left Bousfield localization of this model structure;

• and finally the stabilization may be modeled by further passing to spectra with respect to the suspension operaton $T\mapsto T\wedge X$ in the sense of

  • Mark Hovey, Spectra and symmetric spectra in General Model Categories K-theory, (web, pdf)

  for $T={𝔸}^1/({𝔸}^1-0)$ a suitable model of the circle and $\wedge$ the internal smash product.

A concise overview of the constructions and definitions just outlined above is in

• Rick Jardine, Motivic spaces and the motivic stable category (pdf)

More details on the Nisnevich site, the model structure on simplicial presheaves on it and its homotopy localization to A1-homotopy theory is in section 3 of

• Fabien Morel, Vladimir Voevodsky, ${𝔸}^1$-homotopy theory of schemes , K-theory, 0305 (web pdf)

Grading and bigrading

The (∞,1)-topos $H={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Nis})$ naturally is a lined topos with line object ${𝔸}^1$ that considered in A1-homotopy theory.

Accordingly, by the general reasoning discussed at Cohomology – Bigradings, in $H$ there are two kinds of 1-spheres

• the categorical 1-sphere $S^1={\Delta }^1/\partial {\Delta }^1$, i.e. the $\mathrm{\infty }$-stack obtained by $\mathrm{\infty }$-stackification from the presheaf constant on the groupoid $B\mathbb{Z}=*//\mathbb{Z}$;

• the geometric 1-sphere $S_t^1:={𝔸}^1-\{0\}={𝔾}_m$ (the multiplicative group inside the affine line ${𝔸}^1$);

whose smash product is called

• the Tate sphere $T:={\mathbb{P}}^1:=S^1\wedge S_{{𝔸}^1}^1$ which is equivalent to $\cdots \simeq {𝔸}^1/({𝔸}^1-\{0\})$;

(e.g. MoVo98, p. 79) and which induce a bigrading on cohomology:

$$H^{p,q}(X,A):={\pi }_{0}H(X,{\Omega }_T^{\mathrm{\infty }}{\Sigma }^{p-q}{\Sigma }_t^{q}A).$$H^{p,q}(X, A) := \pi_0 \mathbf{H}(X, \Omega_T^\infty \Sigma^{p-q} \Sigma_{t}^{q} A) \,.

For instance for $A=H(\mathbb{Z})$ the Eilenberg-MacLane object (discussed more in detail below), one writes

$$H^{p,q}(X,\mathbb{Z}):={\pi }_{0}H(X,{\Omega }_T^{\mathrm{\infty }}{\Sigma }^{p-q}{\Sigma }_t^{q}H(\mathbb{Z})).$$H^{p,q}(X, \mathbb{Z}) := \pi_0 \mathbf{H}(X, \Omega_T^\infty \Sigma^{p-q} \Sigma_{t}^{q} H(\mathbb{Z})) \,.

Chow groups as motivic cohomology with coefficients in Eilenberg-MacLane objects

To define an Eilenberg-MacLane object representing an interesting cohomology theory the first guess might be to apply the general definition of an Eilenberg-MacLane object given under the link, which amounts to taking the constant Eilenberg-MacLane-space-valued sheaves. This is no good:

1. Philosophically: It will result in a constant simplicial presheaf on the site of schemes, e.g. not varying with different coefficient rings as input and thus disregarding all the algebro-geometric information of a scheme. But motivic cohomology should talk exactly about this.

2. Technically: The general construction would naturally yield an $S^1$-spectrum, but we want a ${\mathbb{P}}^1$-spectrum. This is not easy: Voevodsky states in his ICM-talk article (on p. 596, see references) that every morphism ${\mathbb{P}}^1\wedge K(\mathbb{Z},n)\to K(\mathbb{Z},n+1)$ is trivial in the ${𝔸}^1$-homotopy category.

Instead one applies a recipe which, when applied to the usual topological spheres produces the (topological) Eilenberg-MacLane spaces, to the algebro-geometric sphere ${\mathbb{P}}^1$:

The Dold-Thom theorem says that in topology the reduced singular homology of a space $X$ can be produced as

$$H_i(X)={\pi }_i({\mathrm{colim}}_N{\mathrm{Sym}}^{N}X{)}^{+}$$H_i(X) = \pi_i (\mathrm{colim}_N\, \mathrm{Sym}^N X)^+

where ${\mathrm{Sym}}^{N}X=(X\times X\times ...\times X)/{\Sigma }_N$ is the free monoid on $X$ and $(-{)}^{+}$ denotes group completion. Inserting the topological n-sphere $S^n$ yields that $({\mathrm{Sym}}^{N}X{)}^{+}$ is an Eilenberg-MacLane space.

The construction makes sense for schemes seen as Set-valued functors, in particular for ${\mathbb{P}}^1$. More generally we want to apply it to smash-products of simplicial sheaves ${\mathbb{P}}^1\wedge \dots \wedge {\mathbb{P}}^1$ and define for $i:Z\to X$ a closed immersion (for us e.g. ${\mathbb{P}}^1\times \mathrm{pt}\cup \mathrm{pt}\times {\mathbb{P}}^1\to {\mathbb{P}}^1\times {\mathbb{P}}^1$)

$${\mathrm{Sym}}^N(X/Z)={\mathrm{Sym}}^N(X)/\cup \mathrm{Im}(X\times \dots \times Z\times \dots \times X)$$\mathrm{Sym}^N (X/Z) = \mathrm{Sym}^N (X)/\cup Im(X \times \ldots \times Z \times \ldots \times X)

We have maps ${\mathrm{Sym}}^N(X)\to {\mathrm{Sym}}^{N+1}(X)$ (lengthening an $N$-letter word by one, attaching the base point) and the colimit over these maps, followed by group completion gives a functor $L(-):=({\mathrm{Sym}}^{\mathrm{\infty }}(-){)}^{+}$. Now one defines the motivic Eilenberg-MacLane spaces $K(\mathbb{Z}(n),2n):=L({\mathbb{P}}^1)$.

These assemble to give the motivic Eilenberg-MacLane spectrum $H(\mathbb{Z}):=(*,K(\mathbb{Z}(1),2),K(\mathbb{Z}(2),4),\dots )$ with bonding maps induced by ${\mathbb{P}}^1\wedge {\mathrm{Sym}}^N({{\mathbb{P}}^1}^{\wedge m})\to {\mathrm{Sym}}^N({{\mathbb{P}}^1}^{\wedge m+1}),(x,\sum x_i)\mapsto \sum (x,x_i)$ (i.e. take the extra ${\mathbb{P}}^1$-point as new coordinate in the bigger $\wedge$-product of ${\mathbb{P}}^1$s).

The bigraded homology and cohomology theories associated to this spectrum are called motivic (co)homology.

This definition works at least over characteristic 0. In general one has to take cycles as described by Denis-Charles Cisinski below. Intuitively the points of $({\mathrm{Sym}}^{\mathrm{\infty }}(X){)}^{+}$ are finite formal sums of points of X, i.e. zero-cycles, which links this story to the functor $L$ described below. In characteristic zero both coincide. The link to Chow groups however only becomes apparent in the cycle description.

the following paragraphs are due to Denis-Charles Cisinski, taken from this MathOverflow thread.

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 2). This works essentially like in Top: in the (model) category of simplicial Nisnevich sheaves (over smooth $k$-schemes), the classifying space of the multiplicative group ${𝔾}_m:={𝔸}^1-\{0\}$ has the ${𝔸}^1$-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 $H^1(X,{𝔾}_m)=\mathrm{Pic}(X)$ reads as

$${\pi }_0{H}_{\mathrm{Nis}}(X,B{𝔾}_m)={\mathrm{Ho}}_{\mathrm{SSh}(\mathrm{Nis})}(X,B{𝔾}_m)=\mathrm{Pic}(X)={\mathrm{CH}}^2(X)$$\pi_0 \mathbf{H}_{Nis}(X,\mathbf{B} \mathbb{G}_m) = Ho_{SSh(Nis)}(X, \mathbf{B} \mathbb{G}_m) = Pic(X) = CH^2(X) \,

where ${\pi }_0{H}_{\mathrm{Nis}}(-,-)=\mathrm{Ho}(\mathrm{SSh}(\mathrm{Nis}))(-,-)$ stands for the hom-set in the homotopy category of $k$-schemes.

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

$${\pi }_{0}H({\Sigma }^{i}X,K(\mathbb{Z}(n)),2n))={\mathrm{Ho}}_{\mathrm{SSh}(\mathrm{Nis})}({\Sigma }^{i}X,K(\mathbb{Z}(n)),2n))=H^{2n-i}(X,\mathbb{Z}(n)).$$\pi_0 \mathbf{H}(\Sigma^i X , K(\mathbb{Z}(n)), 2n)) = Ho_{SSh(Nis)}(\Sigma^i X , K(\mathbb{Z}(n)), 2n)) = H^{2n -i}(X, \mathbb{Z}(n)) \,.

There are several models for $K(\mathbb{Z}(n),2n)$, one of the smallest being constructed as follows. What is explained above is that $K(\mathbb{Z}(1),2)$ is the infinite projective space. $K(\mathbb{Z}(0),0)$ is simply the constant sheaf . For higher $n$, here is the following 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 $Sum{n}_i{Z}_i$ 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 $X$ (we 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 the Eilenberg-MacLane object $K(\mathbb{Z}(n),2n)$ in $H_{\mathrm{Nis}}$ 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 $L(u_i):L(Y)\to L(X)$.

References

The model structure on simplicial presheaves on the Nisnevich site and its homotopy localization to A1-homotopy theory is in

• J. F. Jardine, Motivic spaces and the motivic stable category (pdf)

More details are in section 3 of

• Fabien Morel, Vladimir Voevodsky, ${𝔸}^1$-homotopy theory of schemes , K-theory, 0305 (web pdf)

A discussion of the equivariant cohomology case of motivic cohomology is in

• Ben Williams, Equivariant Motivic Cohomology (pdf)

Voevodsky’s definition

In the mid 1990s Vladimir Voevodsky proposed a concrete definition of motivic cohomology of a smooth scheme $X$ over a field as the hypercohomology of certain complex of sheaves? on the Zariski or etale site of $X$ (an analog of the category of open subsets of a topological space). This complex is called the motivic complex; the existence of such a complex was predicted as part of the so-called Beilinson dream.

Voevodsky gave a concrete definition of the derived category of the hypothetical category of mixed motives?. The category of motives has not only more objects but also locally more morphisms than the category of schemes, and it comes with a functor from an appropriate category of schemes. The morphisms are certain correspondences, and Voevodsky has shown that the motivic complexes are realized as derived hom-complexes in his derived category of mixed motives.

Voevodsky’s proposal has been shown to have most properties that Grothendieck and Beilinson had demanded of the hypothetical cohomology theory, except that to date it hasn’t been shown yet that the cohomology groups vanish in negative degree, as they should.

The motivic complex – a chain complex of sheaves with values in abelian groups – is defined in definition 3.1, page 33 of

• Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lectures in motivic cohomology (web pdf)

Definition

The motivic cohomology of a smooth scheme $X$ is the abelian sheaf cohomology, more specifically hypercohomology, of the motivic complex of sheaves with transfers on the Zariski site. See definition 3,4, page 22.

The analogous definition with the Zariski site structure of $X$ replaced by the etale site $\mathrm{Et}(X)$ is in lecture 10.

As hom-sets of motives

Motivic cohomology computes certain derived hom-sets in the category of motives.

This is discussed in lecture 14 of MaVoWe. See prop 14.16.

References

• Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lectures in motivic cohomology (web pdf)

• Vladimir Voevodsky, A^1-homotopy Theory, Documenta Mathematica, Extra Volume ICM 1998, I, 579-604 ps

Further references

• Eric Friedlander, Algebraic Cycles and algebraic K-theory, II (lecture 6 (pdf))

• J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553

• A. Beĭlinson, R. MacPherson, V. Schechtman, Notes on motivic cohomology, Duke Math. J. 54 (1987), no. 2, 679–710; doi.

• Vladimir Voevodsky, Pierre Deligne, Lectures on motivic cohomology 2000/2001 (web)

For a discussion of the relation betwen motives and motivic cohomology, see for instance section 0.1.8 of

• Spencer Bloch?, Lectures on Mixed Motives (ps)

• Marc Hoyois, On the relation between algebraic cobordism and motivic cohomology (pdf)

• Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes (arXiv:1206.3645)

category: algebraic geometry