Motivic cohomology

General Introduction
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 Ktheory, Motivic homotopy theory
Higher Chow groups, Suslin homology, Homology of schemes, FriedlanderSuslin cohomology, Singular homology of varieties
Motivic cohomology is supposed to be the universal BlochOgus cohomology. See also Motivic cohomology with compact supports, BorelMoore motivic homology, BorelMoore homology, Motivic homology
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
<]]> 
Search results
<]]> 
Online References
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 Ktheory handbook (not online)
Friedlander and Kahn (2 articles) in Ktheory handbook.
Other references
Voevodsky's Seattle lectures. In Voevodsky folder and probably online.
Levine: Mixed motives. Probably, this is the same as K0107
Levine on lambda operations on Ktheory
Something really interesting by May on caterads: here and here
Interesting notes by Friedlander on motivic complexes: K0216
Voevodsky: Motivic cohomology is isomorphic to higher Chow groups.
Geisser: Motivic cohomology over Dedekind rings. Relations to BlochKato and BeilinsonLichtenbaum.
See also all papers by Voevodsky! Also Suslin, Friedlander.
For the theory of sheaves with transfers, there are nice expositions by Déglise, here and in Nagel and Peters vol I.
Beilinson's original letter
A survey by Niziol on "padic motivic cohomology" or motivic cohomology of arithmetic schemes.
<]]> 
Paper References
André: Une introduction aux motifs. My notes
Goncharov in Ktheory handbook, for various motivic complexes.
Lichtenbaum: Motivic complexes (in Motives vol). Read again!
<]]> 
Definition
There is a description of motivic EilenbergMacLane spaces in Voevodsky ICM talk, using the right analogue of the DoldThom construction in topology.
Memo notes from Friedlander: Motivic complexes (Ktheory 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 DoldThom 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 . We have symmetric products , also of finte type over . We let , and let be the corresponding cosimplicial scheme. We define the Suslin complex to be the chain complex associated to the simplicial abelian group . (Plus denotes group completion)
Important aspects of the Suslin complex:
 for a certain Nisnevich sheaf on . This sheaf is actually a presheaf with transfers.
 Write for the complex of Nisnevich sheaves with transfers defined by . The homology presheaves of this complex are homotopy invariant
Theorem: Let X be a quasiprojective scheme over an algebraically closed field , and let be a positive integer relatively prime to the exponential characteristic of . Then the mod cohomology of , i.e. the cohomology of the complex , 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 is a perfect field. Let be a short exact sequence of Nisnevich sheaves with transfers (on ). Then the resulting triple of chain complexes of abelian groups is a distinguished triangle, so determines a long exact sequence of homology groups.
Corollary: Let be a covering of a scheme of finite type over a perfect field . Then
is a d.t.
Now to the definitions. Let be the category of smooth schemes over (so in particular of finite type over ). 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 (schemes of finite type over ). This is the minimal Grothendieck topology for which Nisnevich coverings are coverings, and also proper surjective morphisms of the following type: where is a closed embedding and is an isomorphism.
We say that a field admits resolution of singularities if
 For any there is a proper birational surjective morphism such that is smooth over .
 For any smooth scheme over and any proper, birational, surjective map , there exists a sequence of blowups with smooth centers such that factors through .
We define a presheaf by assigning to the free abelian group on the set of integral closed subschemes on which are finite and surjective over . 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 to the free abelian group on .
We also define the Nisnevich sheaf by sending a connected smooth scheme to the group of cycles on equidimensional of relative dimension over . If is proper over , then .
One reason to consider the Nisnevich and cdh topologies rather than the Zariski topology is the existence of MayerVietoris, localization, and blowup exact sequences. Exact statements, see page 1087 of Friedlander in Ktheory handbook.
Def: : Objects are smooth schemes over , and morphisms are given by , the free abelian group of finite correspondences from to . A presheaf with transfers is a contravariant 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 on , write for the complex of presheaves on sending to the complex .
Lemma: Let be a presheaf on . Consider , the presheaf sending to the th homology of (for some nonnegative integer ). Then is homotopy invariant.
Theorem: (Rigidity thm) Let be a homotopy invariant presheaf with transfers satisfying for some prime to the residue characteristic of . Let be the henselization of affine space at the origin. Then .
Theorem: Let be a homotopy invariant presheaf with transfers. Then its associated Nisnevich sheaf is also a homotopy invariant presheaf with transfers, and equals (as a presheaf on ) the associated Zariski sheaf . Moreover, if is perfect, then
for any , and these are homotopy invariant presheaves with transfer.
Proposition: Assume that is perfect, admitting resolution of singularities. Let be a homotopy invariant presheaf with transfers. Then for any smooth scheme of finite type over , is independent of the topology
The triangulated category
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 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 in this category, one obtains the triangulated category of geometric motives.
Here, we'll focus on another triangulated category of Voevodsky, here denoted by (Voevodsky writes ) The category of effective geometric motives embeds as a full triangulated subcat of . Also, under this embedding the Tate motive is quasiinvertible, so is also a full triangulated subcategory of .
Definition: Let be a scheme over a field . Assume either smooth, or of finite type over a field addmitting RoS. Define the motive of to be
We also define the motive of withe compact supports by
Remark on conventions for complexes:
Next: The triangulated category , designed to capture the Nisnevich cohomology of smooth schemes over , and the cdh cohomology of schemes of finite type over .
Write for the category of Nisnevich sheaves with transfers, and consider its derived category (D^{}, bounded above). Define 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 smooth, we have for any ,
In particular, for smooth,
If is of finite type over , with admitting RoS, then
Corollary: If is of finite type over , then the homology of is given by .
Theorem: Assume is a presheaf with transfers on where is perfect and admitting RoS. If , then is quasiisomorphic to .
Corollary: Some distinguished triangles for motives (MayerVietoris, localization, blowups)
Def: Tate motive, notation , is the cone of . Also Tate twist: . 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 whose cohomology and homology is motivic cohomology and homology. Slightly complicated definition. If admits RoS, this becomes:
In particular, we get (the constant sheaf) and
Now let be a scheme of finite type over the field . We define motivic cohomology of by
For a positive integer , define mod motivic cohomology by:
Previous results imply that for smooth and perfect, motivic cohomology is Zariski hypercohomology:
The last term here is also valid when admits RoS, for any of finite type over .
Fact: Write for . Then motivic cohomology vanishes for .
Theorem: For any field , and any nonnegative integer , 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 be schemes of finite type over a field admitting RoS. Then the natural map
is an isomorphism.
<]]> 
Properties
<]]> 
Standard theorems
Voevoedsky: Cancellation theorem, file in Voevodsky folder, Jan 2002. Proves that if k is a perfect field, than for any in , the map 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 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 cohomology 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, MayerVietoris and Gysin long exact sequences, projective bundle thm, blowup triangles, and a list of comparison results, see p vi.
http://mathoverflow.net/questions/28856/themotiviccohomologyofprojectivespace
<]]> 
Open Problems
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 Ktheory over finite fields and its implications for motivic cohomology with compact support.
BeilinsonLichtenbaum conjecture
The BeilinsonLichtenbaum conjecture predicts that the conjectural map of spectral sequences from the "motivic" spectral sequence (motivic cohomology to algebraic Ktheory mod) to the AtiyahHirzebruch spectral sequence (etale cohomology to etale Ktheory mod) should be an isomorphism on terms (except for a fringe effect related to mod etale cohomological dimension of ).
The point of this would be to reduce the computation of mod algebraic Ktheory to of smooth schemes to topological invariants which in many cases are known.
The BlochKato conjecture for a field and a prime (invertible in ) implies the BeilinsonLichtenbaum conjecture for and .
This has lead to (Kahn, Weibel, Rognes) computations of the 2primary part of rings of the Ktheory of rings of integers in number fields.
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).
<]]> 
Connections to Number Theory
For motivic cohomology over arithmetic schemes, Annette Huber said various things, maybe most importantly that there is something by Faltings defining the motivic cohomology 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 "fmotivic cohomology". All this should be related to Scholbach's motives over Z.
http://londonnumbertheory.wordpress.com/2010/02/25/toaddtothestudygrouptalkyesterday/
<]]> 
Computations and Examples
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 K0331
A computation by Walker
<]]> 
History and Applications
Some kind of application by Kahn
<]]> 
Some Research Articles
Voevodsky: On the zero slice of the sphere spectrum. Talks among other things about the slice filtration and about a certain model for the EM 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: BlochKato conj and motivic cohomology 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 Ktheory and motivic cohomology (1995)
Suslin: On the Grayson spectral sequence (preprint 2002)
Suslin, Voevodsky: BlochKato 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_2terms 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 BlochKato conjecture holds. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the GeisserLevine 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."
Ktheory and motivic cohomology of schemes, by Marc Levine: K0336
Something on BlochKato, by Geisser and Levine
Wildeshaus on the Eisenstein symbol
Levine on inverting the Bott element
Something (wrong) by Walker
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 Déglise: K0764, K0765, K0766
The Gersten conjecture for Milnor Ktheory , by Moritz Kerz, implying Levine's generalized BlochKato conjecture
Bigraded equivariant cohomology of real quadrics , by Pedro F. dosSantos and Paulo LimaFilho: K0785
Walker thesis (1996): Motivic complexes and the Ktheory of automorphisms. (Available on microfilm!)
MR1265532 (95h:19001) Bloch, Spencer(1CHI) An elementary presentation for $K$groups and motivic cohomology. Motives (Seattle, WA, 1991), 239244, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
MR1715883 (2000i:14013) Collino, Alberto(ITRIN) Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds. (English summary) Algebraic $K$theory and its applications (Trieste, 1997), 370402, World Sci. Publ., River Edge, NJ, 1999.
MR2266890 (2007j:11153) Hornbostel, Jens(DRGBG); Kings, Guido(DRGBG) On noncommutative 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 Ktheory
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 for the anisotropic quadric over the field of real numbers.
Levine and Geisser: The Ktheory of fields in char .
<]]> 
Other Information
Bloch: Think of a correspondence as
See this MO question for two different versions of motivic cohomology.
<]]> 
Comments Posted
<]]> 
Comments
There are no comments.