Holmstrom Motivic cohomology

Motivic cohomology

Some kind of application by Kahn

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Motivic cohomology

For products, see Weibel: http://www.math.uiuc.edu/K-theory/0263. Probably same as Weibel: Products in higher Chow groups and motivic cohomology (1999)

Higher Chern Classes and Steenrod Operations in Motivic Cohomology, by Oleg Pushin: In this paper we construct the higher Chern classes in motivic cohomology by adopting the well-known approach due to H. Gillet. We compute, under certain assumption, the composite of a motivic reduced power operation with such a class. We use the higher Chern classes to express the motivic algebra of a general linear groupscheme.

Reduced power operations in motivic cohomology, by Vladimir Voevodsky: http://www.math.uiuc.edu/K-theory/0487

Nie on motivic cohomology operations.

Karoubi’s construction for motivic cohomology operations, by Zhaohu Nie: We use an analogue of Karoubi’s construction in the motivic situation to give some cohomology operations in motivic cohomology. We prove many properties of these operations, and we show that they coincide, up to some nonzero constants, with the reduced power operations in motivic cohomology originally constructed by Voevodsky. The relation of our construction to Voevodsky’s is, roughly speaking, that of a fixed point set to its associated homotopy fixed point set.

Beilinson defined a $\mathbb{Q}$-vector subspace of rational motivic cohomology, called the integral part, for a projective regular variety $X$ over a number field. Scholl extended this to regular projective schemes over a number field, using alterations. It coincides with the image of the rational motivic cohomology of a regular, flat over $Spec(\mathbb{Z})$, projective model.

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Motivic cohomology

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Motivic cohomology

AG (Algebraic geometry)

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Motivic cohomology

Mixed

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Motivic cohomology

André: Une introduction aux motifs. My notes

Goncharov in K-theory handbook, for various motivic complexes.

Lichtenbaum: Motivic complexes (in Motives vol). Read again!

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Motivic cohomology

Asked Jakob Scholbach whether the def of motivic cohomology is independent of the base, see his thesis for background. Answer: Yes, this is independent. Intuitively, you should think that K-theory is something independent of the base. Formally, this is by adjunction: $M_S(X) := f_* 1$, where $f: X \rightarrow S$ is the structural map. Let $g: S \rightarrow T$ be a map of base schemes. Then g_* M_S(X) = M_T(X) Hom_S(1, M_S(X)) = Hom_S(1, f_* 1) = Hom_X(f^* 1, 1) = Hom_X(1, 1) and similarly over T. In somewhat loose accordance to that notice that L-functions are also independent of the base (Lemma 5.1.3).

See Kahn survey for blabla

Check Jannsen and Saito for finiteness results

http://www.math.uiuc.edu/K-theory/0379: “For this we use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic.”

The Motivic DGA, by Roy Joshua: http://www.math.uiuc.edu/K-theory/0470

Équivalence rationnelle, équivalence numérique et produits de courbes elliptiques sur un corps fini, by Bruno Kahn http://www.math.uiuc.edu/K-theory/0585

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Motivic cohomology

Coincides with higher Chow groups, for smooth schemes over a field of char zero.

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Motivic cohomology

There is a description of motivic Eilenberg-MacLane spaces in Voevodsky ICM talk, using the right analogue of the Dold-Thom construction in topology.

Memo notes from Friedlander: Motivic complexes (K-theory handbook)

Beilinson predicted that motivic cohomology should arise as the cohomology of certain complexes. There are several different suggestions for such complexes.

Suslin complexes (“algebraic singular complexes???”)

Motivation: a theorem of Dold-Thom describing the singular cohomology of a topological space group completion of a simplicial abelian monoid built from symmetric products of the space.

Start with a scheme of finite type over a field $k$. We have symmetric products $SP^d(X)$, also of finte type over $k$. We let $\Delta^n = Spec k[t_0, \ldots t_n] / \sum t_i - 1$, and let $\Delta^*$ be the corresponding cosimplicial scheme. We define the Suslin complex $Sus_*(X)$ to be the chain complex associated to the simplicial abelian group $\big( \coprod_d Hom_{Sch/k} (\Delta^*, SP^d(X) ) \big)^{+}$. (Plus denotes group completion)

Important aspects of the Suslin complex:

• $Sus_*(X) = c_{equi}(X,0) (\Delta^*)$ for a certain Nisnevich sheaf $c_{equi}(X,0)$ on $Sm/k$. This sheaf is actually a presheaf with transfers.
• Write $C_*(c_{equi}(X,0) )$ for the complex of Nisnevich sheaves with transfers defined by $U \mapsto c_{equi}(X,0) (U \times \Delta^*)$. The homology presheaves of this complex are homotopy invariant

Theorem: Let X be a quasi-projective scheme over an algebraically closed field $k$, and let $n$ be a positive integer relatively prime to the exponential characteristic of $k$. Then the mod $n$ cohomology of $Sus_*(X)$, i.e. the cohomology of the complex $RHom(Sus_*(X), \mathbb{Z}/n)$, is given by

Proof: Uses rigidity theorem of Suslin and Voevodsky, and a comparison between cohomology in the qfh topology and the etale topology.

Nisnevich sheaves with transfers

A motivating fact:

Theorem: Assume $k$ is a perfect field. Let $0 \to F_1 \to F_2 \to F_3 \to 0$ be a short exact sequence of Nisnevich sheaves with transfers (on $Sm/k$). Then the resulting triple of chain complexes of abelian groups $F_1(\Delta^*) \to F_2(\Delta^*) \to F_3(\Delta^*) \to F_1(\Delta^*)[1]$ is a distinguished triangle, so determines a long exact sequence of homology groups.

Corollary: Let $X = U \cup V$ be a covering of a scheme of finite type over a perfect field $k$. Then

is a d.t.

Now to the definitions. Let $Sm/k$ be the category of smooth schemes over $k$ (so in particular of finite type over $k$). We have the Nisnevich topology on this category. One key property: Points are Hensel local rings.

We want to talk about singular schemes which admit resolutions by smooth schemes. For this purpose we introduce the stronger cdh topology on $Sch/k$ (schemes of finite type over $k$). This is the minimal Grothendieck topology for which Nisnevich coverings are coverings, and also proper surjective morphisms of the following type: $W \coprod U_1 \to U$ where $U_1 \to U$ is a closed embedding and $p^{-1}(U-U_1) \to (U-U_1)$ is an isomorphism.

We say that a field admits resolution of singularities if

1. For any $X \in Sch/k$ there is a proper birational surjective morphism $Y \to X$ such that $Y$ is smooth over $k$.
2. For any smooth scheme $X$ over $k$ and any proper, birational, surjective map $q: X' \to X$, there exists a sequence of blowups $p: X_n \to \ldots \to X_1 = X$ with smooth centers such that $p$ factors through $q$.

We define a presheaf $c_{equi}(X,0)$ by assigning to $U \in Sm/k$ the free abelian group on the set of integral closed subschemes on $X \times U$ which are finite and surjective over $U$. It is a sheaf for the etale and the Nisnevich topologies. It can be constructed as a sheaf even for the qfh topology, by taking the sheaf associated to the presheaf sending $U$ to the free abelian group on $Hom_{Sch/k}(U, X)$.

We also define the Nisnevich sheaf $z_{equi}(X,r)$ by sending a connected smooth scheme $U$ to the group of cycles on $U \times X$ equidimensional of relative dimension $r$ over $U$. If $X$ is proper over $k$, then $c_{equi}(X,0) = z_{equi}(X,0)$.

One reason to consider the Nisnevich and cdh topologies rather than the Zariski topology is the existence of Mayer-Vietoris, localization, and blow-up exact sequences. Exact statements, see page 1087 of Friedlander in K-theory handbook.

Def: $SmCor(k)$: Objects are smooth schemes over $k$, and morphisms are given by $Hom(U,X) = c_{equi}(X,0)(U)$, the free abelian group of finite correspondences from $U$ to $X$. A presheaf with transfers is a contravariant $Ab$-valued functor on this category.

Note: Earlier papers use condition “pretheory of homological type” on a presheaf. This condition is implied by existence of transfers.

Notation: For a presheaf $F$ on $Sm/k$, write $\underline{C}_*(F)$ for the complex of presheaves on $Sm/k$ sending $U$ to the complex $F(U \times \Delta^*)$.

Lemma: Let $F$ be a presheaf on $Sm/k$. Consider $h^{-i}(F)$, the presheaf sending $U$ to the $i$-th homology of $\underline{C}_*(F)$ (for some non-negative integer $i$). Then $h^{-i}(F)$ is homotopy invariant.

Theorem: (Rigidity thm) Let $\Psi$ be a homotopy invariant presheaf with transfers satisfying $n \Psi = 0$ for some $n$ prime to the residue characteristic of $k$. Let $S_d$ be the henselization of affine $d$-space at the origin. Then $\Psi(S_d) = \Psi(Spec(k))$.

Theorem: Let $F$ be a homotopy invariant presheaf with transfers. Then its associated Nisnevich sheaf $F_{Nis}$ is also a homotopy invariant presheaf with transfers, and equals (as a presheaf on $Sm/k$) the associated Zariski sheaf $F_{Zar}$. Moreover, if $k$ is perfect, then

for any $i \geq 0$, and these are homotopy invariant presheaves with transfer.

Proposition: Assume that $k$ is perfect, admitting resolution of singularities. Let $F$ be a homotopy invariant presheaf with transfers. Then for any smooth scheme of finite type over $k$, $H^*_{?}(X, F_?)$ is independent of the topology $? \in \{Nis, Zar, cdh \}$

The triangulated category $DM_k$

In Voevodsky’s approach to motives for smooth schemes and for schemes of finite type over a field admitting resolution of singularities, there is a triangulated category $DM^{eff}_{gm}(k)$ of effective geometric motives. Roughly speaking, this category is by adjoining kernels and cokernels of projectors to the localization of the homotopy category of bounded complexes on the category of smooth schemes and finite correspondences. Inverting the Tate object $\mathbb{Z}(1)$ in this category, one obtains the triangulated category $DM_{gm}(k)$ of geometric motives.

Here, we’ll focus on another triangulated category of Voevodsky, here denoted by $DM_k$ (Voevodsky writes $DM^{eff}_{-}(k)$) The category of effective geometric motives embeds as a full triangulated subcat of $DM_k$. Also, under this embedding the Tate motive is quasi-invertible, so $DM_{gm}$ is also a full triangulated subcategory of $DM_k$.

Definition: Let $X$ be a scheme over a field $k$. Assume either $X$ smooth, or $X$ of finite type over a field addmitting RoS. Define the motive of $X$ to be

We also define the motive of $X$ withe compact supports by

Remark on conventions for complexes: $H^i(X, K[1]) = H^{i+1}(X,k)$

Next: The triangulated category $DM_k$, designed to capture the Nisnevich cohomology of smooth schemes over $k$, and the cdh cohomology of schemes of finite type over $k$.

Write $Shv_{Nis}(SmCor(k))$ for the category of Nisnevich sheaves with transfers, and consider its derived category (D^{-}, bounded above). Define $DM_k$ as the subcategory of this derived category consisting of complexes with homotopy invariant cohomology sheaves. Note that the above defined motives are members of this category.

Proposition: For $X$ smooth, we have for any $K \in DM_k$,

In particular, for $X$ smooth,

If $X$ is of finite type over $k$, with $k$ admitting RoS, then

Corollary: If $Y$ is of finite type over $k$, then the homology of $Sus_*(Y)$ is given by $Hom_{DM_k}(\mathbb{Z}[*], M(Y))$.

Theorem: Assume $F$ is a presheaf with transfers on $Sm/k$ where $k$ is perfect and admitting RoS. If $F_{cdh} = 0$, then $\underline{C}_*(F)_{Zar}$ is quasi-isomorphic to $0$.

Corollary: Some distinguished triangles for motives (Mayer-Vietoris, localization, blow-ups)

Def: Tate motive, notation $\mathbb{Z}(1)[2]$, is the cone of $M(Spec(k)) \to M(\mathbb{P}^1)$. Also Tate twist: $M(X)(1) = cone \big( M(X) \to M(X \times \mathbb{P}^1)[-2] \big)$. Same for motives with compact supports.

Remark: One can replace the Nisnevich topology above with the etale topology, and express etale cohomology as a Hom in the corresponding DM category, just as above.

Motivic cohomology and homology

We now define the motivic complexes $\mathbb{Z}(n) \in DM_k$ whose cohomology and homology is motivic cohomology and homology. Slightly complicated definition. If $k$ admits RoS, this becomes:

In particular, we get $\mathbb{Z}(0) = \mathbb{Z}$ (the constant sheaf) and $\mathbb{Z}(1)[1] = \mathbb{G}_m$

Now let $X$ be a scheme of finite type over the field $k$. We define motivic cohomology of $X$ by

For a positive integer $m$, define mod $m$ motivic cohomology by:

Previous results imply that for $X$ smooth and $k$ perfect, motivic cohomology is Zariski hypercohomology:

The last term here is also valid when $k$ admits RoS, for any $X$ of finite type over $k$.

Fact: Write $d$ for $dim(X)$. Then motivic cohomology vanishes for $i > d+j$.

Theorem: For any field $k$, and any non-negative integer $n$, there is a natural isomorphism:

Hence motivic cohomology is defined using the Zariski site, for smooth schemes, and the cdh topology for more general schemes. Using the etale topology instead, one obtains etale motivic cohomology.

Def of motivic cohomology with compact supports, motivic homology, and motivic homology with locally compact supports. Duality and relation to bivariant theory.

Theorem: There is a natural isomorphism with Bloch’s higher Chow groups:

Theorem: Let $X,Y$ be schemes of finite type over a field $k$ admitting RoS. Then the natural map

is an isomorphism.

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Motivic cohomology

Motivic cohomology, first envisioned by Beilinson (I think) and then developed by Voevodsky and others in the 90s, has been the subject of much interesting progress. For the material on this page, we are very indebted to sources like Friedlander’s Bourbaki article and the book by Mazza/Voevodsky/Weibel.

See also:

Mixed motives, Voevodsky motives, Algebraic K-theory, Motivic homotopy theory

Higher Chow groups, Suslin homology, Homology of schemes, Friedlander-Suslin cohomology, Singular homology of varieties

Motivic cohomology is supposed to be the universal Bloch-Ogus cohomology. See also Motivic cohomology with compact supports, Borel-Moore motivic homology, Borel-Moore homology, Motivic homology

Etale motivic cohomology

Motivic cohomology is representable in the motivic stable homotopy category; see Motivic homotopy theory.

Motivic cohomology can be viewed as an Absolute cohomology theory.

It is also an Oriented cohomology theory.

Possible the following notions are of some relevance (but maybe not): Pretheory, Mixed Weil cohomology, Twisted duality theory

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Motivic cohomology

Voevodsky’s Nordfjordeid lecture - supernice (Voevodsky folder). Working over a field unfortunately. Basic constructions, of SH etc. Brief discussion of Thom spaces and homotopy purity. Cohomology theories: the motivic EM spectrum, KGL, MGL, claim that the notions of orientation and FGLs have direct analogs for P1-spectra. The slice filtration (great intro), update on Open problems paper. The zero-th slice of the unit spectrum is HZ, this is known for fields of char zero. Brief discussion of AHSS. Appendix on the Nisnevich topology, Nisnevich descent, and model structures.

Voevodsky: Motivic Eilenberg-MacLane spaces. We consider a perfect field k and an admissible subcat of $Sch/k$. For such a subcat we have two homotopy cats,namely the homotopy theory version, and the correspondence version (depending on a choice of coefficient ring). Functors between these cats. Def of EM spaces in the homotopy theory setting. Under some assumptions, we can say something about the image of an EM space in the correspondence setting: namely that for $p \geq q$, we get a mixed Tate object, and for $p \geq 2q$ a pure Tate object. This “should in principle be sufficient to show that all bi-stable operations are obtained from reduced powers and Bockstein”, but some problems have not been worked out completely. For $k$ of char zero, we can give a description of the motivic Steenrod algebra. Remark: We know almost nothing about EM spaces for $pq$ and $q>1$, this is related to the vision of an abelian cat of mixed motives, with a weight filtration. Ref to Sasha letter. More info: Lemma 4.36 on p 64, says that there is a canonical iso from the topological realization of $K(A,p,q)$ to $K(A,p)$ when $p \geq q$, but this fails for $pq$ at least for non-torsion coefficients. From this we get maps from the motivic cohomology of $X$ to the p-th singular cohomology of its complex realization.

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Motivic cohomology

http://mathoverflow.net/questions/6834/kunneth-formula-for-motivic-cohomology

http://mathoverflow.net/questions/55950/is-there-a-universal-coefficient-theorem-for-motivic-cohomology

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Motivic cohomology

arXiv:1002.0105 Parshin’s conjecture and motivic cohomology with compact support from arXiv Front: math.KT by T. Geisser We discuss Parshin’s conjecture on rational K-theory over finite fields and its implications for motivic cohomology with compact support.

Beilinson-Lichtenbaum conjecture

The Beilinson-Lichtenbaum conjecture predicts that the conjectural map of spectral sequences from the “motivic” spectral sequence (motivic cohomology to algebraic K-theory mod-$\ell$) to the Atiyah-Hirzebruch spectral sequence (etale cohomology to etale K-theory mod-$\ell$) should be an isomorphism on $E_2$-terms (except for a fringe effect related to mod $\ell$ etale cohomological dimension of $X$).

The point of this would be to reduce the computation of mod $\ell$ algebraic K-theory to of smooth schemes to topological invariants which in many cases are known.

The Bloch-Kato conjecture for a field $k$ and a prime $\ell$ (invertible in $k$) implies the Beilinson-Lichtenbaum conjecture for $k$ and $\ell$.

This has lead to (Kahn, Weibel, Rognes) computations of the 2-primary part of rings of the K-theory of rings of integers in number fields.

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Nekovar in Motives, page 558: Beilinson conjectures that rational motivic cohomology (and the regulator) extends to the cat of motives mod homological equivalence. (As shown, it does extend to Chow motives).

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Motivic cohomology

For motivic cohomology over arithmetic schemes, Annette Huber said various things, maybe most importantly that there is something by Faltings defining the motivic cohom of a scheme over say Z as the image in the thing over Q under that map induced by a model. This should give independence of model I think. Ref to a Bourbaki talk, maybe by Faltings. Also ref to “f-motivic cohomology”. All this should be related to Scholbach’s motives over Z.

http://londonnumbertheory.wordpress.com/2010/02/25/to-add-to-the-study-group-talk-yesterday/

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Motivic cohomology

Voevodsky: On the zero slice of the sphere spectrum. Talks among other things about the slice filtration and about a certain model for the E-M spectrum, which only is valid over a field of characteristic zero.

arXiv:1207.3121 Opérations de Steenrod motiviques fra arXiv Front: math.AT av Joël Riou This article fills some gaps in Voevodsky’s construction of the Steenrod operations acting on the motivic cohomology with coefficients in Z/lZ of motivic spaces in the sense of Morel and Voevodsky over a perfect field of characteristic different from l. Moreover, as a consequence of the method of proof of a theorem by Voevodsky on stable cohomology operations, we show that the spectrum that represents motivic cohomology with coefficients Z/lZ has no nonzero “superphantom” endomorphism.

Voevodsky and Suslin: Bloch-Kato conj and motivic cohom with finite coeffs, file in Voevodsky folder. Contains lots of material on Nisnevich sheaves with transfers and other foundational matters.

Shuji Saito research summary, he has written on motivic cohomology of arithmetic schemes, in particular on finiteness results

Suslin: Algebraic K-theory and motivic cohomology (1995)

Suslin: On the Grayson spectral sequence (preprint 2002)

Suslin, Voevodsky: Bloch-Kato conjecture and motivic cohomology with finite coefficients (2000)

Kahn and Huber: In this paper we study the “slice filtration” defined by effectivity conditions on Voevodsky’s triangulated motives, and apply it to obtain spectral sequences converging to their motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their E_2-terms then have a simple description. We apply this in particular to get spectral sequences converging to the motivic cohomology of a split connected reductive group.

Kahn: We reformulate part of the arguments of T. Geisser and M. Levine computing motivic cohomology with finite coefficients under the assumption that the Bloch-Kato conjecture holds. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories.

A paper by Scholl: “In the first part of this paper we use de Jong’s method of alterations to contruct unconditionally the `integral' subspaces of motivic cohomology (with rational coefficients) for Chow motives over local and global fields - for the motive of a smooth and proper variety possessing a regular model over the ring of integers of the base field, this coincides with the image of the motivic cohomology of the model in that of the generic fibre."

K-theory and motivic cohomology of schemes, by Marc Levine: http://www.math.uiuc.edu/K-theory/0336

Something on Bloch-Kato, by Geisser and Levine

Wildeshaus on the Eisenstein symbol

Levine on inverting the Bott element

Something (wrong) by Walker

Yagita

Some characteristic p thing by Levine and Geisser

Coniveau spectral sequence and motivic cohomology of quadrics and classifying spaces, by Nobuaki Yagita

Some work by Deglise: http://www.math.uiuc.edu/K-theory/0764, http://www.math.uiuc.edu/K-theory/0765, http://www.math.uiuc.edu/K-theory/0766

The Gersten conjecture for Milnor K-theory , by Moritz Kerz, implying Levine’s generalized Bloch-Kato conjecture

Bigraded equivariant cohomology of real quadrics , by Pedro F. dosSantos and Paulo Lima-Filho: http://www.math.uiuc.edu/K-theory/0785

Walker thesis (1996): Motivic complexes and the K-theory of automorphisms. (Available on microfilm!)

MR1265532 (95h:19001) Bloch, Spencer(1-CHI) An elementary presentation for $K$-groups and motivic cohomology. Motives (Seattle, WA, 1991), 239–244, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

MR1715883 (2000i:14013) Collino, Alberto(I-TRIN) Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds. (English summary) Algebraic $K$-theory and its applications (Trieste, 1997), 370–402, World Sci. Publ., River Edge, NJ, 1999.

MR2266890 (2007j:11153) Hornbostel, Jens(D-RGBG); Kings, Guido(D-RGBG) On non-commutative twisting in étale and motivic cohomology.

Geisser: Motivic cohomology over Dedekind rings (2004)

Nisnevich: The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory

Yagita: Note on the mod p motivic cohomology of algebraic groups.

Yagita: Note on motivic cohomology of anisotropic real quadrics. Abstract: In this paper, we compute the mod 2 motivic cohomology $H^{*,*'}(X;Z/2)$ for the anisotropic quadric $X$ over $R$ the field of real numbers.

Levine and Geisser: The K-theory of fields in char $p$.

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Motivic cohomology

Favourite references:

Mazza, Voevodsky, Weibel: Lectures in motivic cohomology. Most of MVW treats only the case where the base scheme is a field. We have not really taken any notes from MVW in the DB.

Here is what I guess is the orange book, by Friedlander, Suslin and Voevodsky.

Geisser in K-theory handbook (not online)

Friedlander and Kahn (2 articles) in K-theory handbook.

Other references

Voevodsky’s Seattle lectures. In Voevodsky folder and probably online.

Levine: Mixed motives. Probably, this is the same as http://www.math.uiuc.edu/K-theory/0107

Levine on lambda operations on K-theory

Something really interesting by May on caterads: here and here

Interesting notes by Friedlander on motivic complexes: http://www.math.uiuc.edu/K-theory/0216

Voevodsky: Motivic cohomology is isomorphic to higher Chow groups.

Geisser: Motivic cohomology over Dedekind rings. Relations to Bloch-Kato and Beilinson-Lichtenbaum.

See also all papers by Voevodsky! Also Suslin, Friedlander.

For the theory of sheaves with transfers, there are nice expositions by Deglise, here and in Nagel and Peters vol I.

Beilinson’s original letter

A survey by Niziol on “p-adic motivic cohomology” or motivic cohomology of arithmetic schemes.

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Motivic cohomology

Motivic cohomology of smooth cellular varieties, by Kahn

Ben Williams computes motivic cohomology of Stiefel varieties, in some paper (2011).

Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces , by Alexander Nenashev and Kirill Zainoulline

Motivic cohomology of linear varieties: Joshua, see also http://www.math.uiuc.edu/K-theory/0331

A computation by Walker

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Motivic cohomology

Voevoedsky: Cancellation theorem, file in Voevodsky folder, Jan 2002. Proves that if k is a perfect field, than for any $K, L$ in $DM^{eff}_{-}(k)$, the map $Hom(K,L) \to Hom(K(1), L(1))$ is bijective. This was earlier known when k admits RoS (in Triangulated cats of motives over a field), and also for any field in the case where $L$ is a shift of a motivic complex (in Motivic cohomology are isomorphic to higher Chow groups).

Voevodsky: Motivic cohomology are isomorphic to higher Chow groups. Shows that two defs (MVW version and SF version I think)of motivic cohom agree, and that they agree with higher Chow groups for smooth varieties, and that the Cancellation theorem holds, both statements for any base field, without RoS assumption. A nice discussion of the different versions of motivic complexes.

From MVW intro: Homotopy invariance, Mayer-Vietoris and Gysin long exact sequences, projective bundle thm, blowup triangles, and a list of comparison results, see p vi.

http://mathoverflow.net/questions/28856/the-motivic-cohomology-of-projective-space

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Motivic cohomology

Bloch: Think of a correspondence as $U \to Sym^n T$

See this MO question for two different versions of motivic cohom.

http://mathoverflow.net/questions/102839/what-is-the-relationship-between-motivic-cohomology-and-the-theory-of-motives

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Motivic cohomology

[CDATAThere are (at least) 2 ss converging to motivic cohomology: Bloch-Ogus, or coniveau ss, and also the slice ss. (I think Kahn said this in a talk)]

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In topology, one has the Atiyah-Hirzebruch spectral sequence: $E^{p,q}_2 = H^p(T, K^q_{top}) \implies K^{p+q}_{top}(T)$. Here $K^q_{top} = \mathbb{Z}$ for even negative $q$, and zero otherwise. When tensored with the rationals, this spectral sequence collapses and yields

This decomposition coincides with the weight spaces of the Adams operations.

Grothendieck proved, for a smooth scheme $X$, the formula:

which is also given by the Adams operation weight spaces.

Dwyer and Friedlander developed etale K-theory for schemes, and there is an Atiyah-Hirzebruch spectral sequence from etale cohomology to etale K-theory.

Bloch introduced complexes $Z_*^d$ for $X$ quasi-projective over a field. Write $CH^d(X, n)$ for the n-th homology of this complex. For a smooth scheme, we have

this composition being "presumably" given by Adams operation weight spaces.

Bloch and Lichtenbaum established a spectral sequence for the spectrum of a field: $E^{p,q}_2 = CH^{-q}(F, -p-q) \implies K_{-p-q}(F)$. Beilinson anticipated a long time ago that there should be such a spectral sequence for quite a general kind of smooth scheme:

The $E_2$-term here is motivic cohomology. Beilinson and Lichtenbaum predicted that this cohomology should be the cohomology of motivic chain complexes. The Suslin-Voevodsky complexes $\mathbb{Z}(n)$ give a motivic cohomology which satisfies most of the expacted properties. Results by Suslin, Friedlander, Voevodsky shows that Bloch’s higher Chow groups $CH^d(X,n)$ equals motivic cohomology for smooth schemes over a field which admits resolution of singularities.

Some references

See the relevant chapter in K-theory handbook!

See Bloch and Lichtenbaum, for a spectral sequence from the motivic cohomology of a field F to its algebraic K-theory. A generalization by Levine, and an alternative approach by Grayson/Suslin

The spectral sequence relating algebraic K-theory to motivic cohomology, by Eric M. Friedlander and Andrei Suslin: http://www.math.uiuc.edu/K-theory/0432

A possible new approach to the motivic spectral sequence, by Vladimir Voevodsky: http://www.math.uiuc.edu/K-theory/0469. From Jan 2001. Says that we do still not have a simple construction of the ss relating mot cohom and alg Kth. Grayson construction simple and elegant but cannot identify the E2-terms with mot cohom. Bloch-Licht-Friedl-Suslin is technically and conceptually very involved. We suggested an approach using slices in open problems paper. For our setting, the problem becomes: relate the slices of K-th with motivic cohom, more precisely $s_0(KGL) = H_Z$ suffices. We show in this paper that two general conjectures about the motivic stable homotopy cat implies the relevant statement about slices of the K-th spectrum. The first of these is the the zero-th slice of the sphere spectrum is the motivic cohom spectrum.

Apparently an application

http://mathoverflow.net/questions/87257/motivic-cohomology-vs-k-theory-for-singular-varieties

nLab page on Motivic cohomology