Motivic cohomology

Last updated December 30, 2023

Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.

Motivic homology and cohomology

Let X be a scheme of finite type over a field k. A key goal of algebraic geometry is to compute the Chow groups of X, because they give strong information about all subvarieties of X. The Chow groups of X have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme Z of X, there is an exact sequence of Chow groups, the localization sequence

CH_{i}(Z)\rightarrow CH_{i}(X)\rightarrow CH_{i}(X-Z)\rightarrow 0,

whereas in topology this would be part of a long exact sequence.

This problem was resolved by generalizing Chow groups to a bigraded family of groups, (Borel–Moore) motivic homology groups (which were first called higher Chow groups by Bloch).^[1] Namely, for every scheme X of finite type over a field k and integers i and j, we have an abelian group H_i(X,Z(j)), with the usual Chow group being the special case

CH_{i}(X)\cong H_{2i}(X,\mathbf {Z} (i)).

For a closed subscheme Z of a scheme X, there is a long exact localization sequence for motivic homology groups, ending with the localization sequence for Chow groups:

\cdots \rightarrow H_{2i+1}(X-Z,\mathbf {Z} (i))\rightarrow H_{2i}(Z,\mathbf {Z} (i))\rightarrow H_{2i}(X,\mathbf {Z} (i))\rightarrow H_{2i}(X-Z,\mathbf {Z} (i))\rightarrow 0.

In fact, this is one of a family of four theories constructed by Voevodsky: motivic cohomology, motivic cohomology with compact support, Borel-Moore motivic homology (as above), and motivic homology with compact support.^[2] These theories have many of the formal properties of the corresponding theories in topology. For example, the motivic cohomology groups Hⁱ(X,Z(j)) form a bigraded ring for every scheme X of finite type over a field. When X is smooth of dimension n over k, there is a Poincare duality isomorphism

H^{i}(X,\mathbf {Z} (j))\cong H_{2n-i}(X,\mathbf {Z} (n-j)).

In particular, the Chow group CHⁱ(X) of codimension-i cycles is isomorphic to H²ⁱ(X,Z(i)) when X is smooth over k.

The motivic cohomology Hⁱ(X, Z(j)) of a smooth scheme X over k is the cohomology of X in the Zariski topology with coefficients in a certain complex of sheaves Z(j) on X. (Some properties are easier to prove using the Nisnevich topology, but this gives the same motivic cohomology groups.^[3]) For example, Z(j) is zero for j < 0, Z(0) is the constant sheaf Z, and Z(1) is isomorphic in the derived category of X to G_m[−1].^[4] Here G_m (the multiplicative group) denotes the sheaf of invertible regular functions, and the shift [−1] means that this sheaf is viewed as a complex in degree 1.

The four versions of motivic homology and cohomology can be defined with coefficients in any abelian group. The theories with different coefficients are related by the universal coefficient theorem, as in topology.

Relations to other cohomology theories

Relation to K-theory

By Bloch, Lichtenbaum, Friedlander, Suslin, and Levine, there is a spectral sequence from motivic cohomology to algebraic K-theory for every smooth scheme X over a field, analogous to the Atiyah-Hirzebruch spectral sequence in topology:

E_{2}^{pq}=H^{p}(X,\mathbf {Z} (-q/2))\Rightarrow K_{-p-q}(X).

As in topology, the spectral sequence degenerates after tensoring with the rationals.^[5] For arbitrary schemes of finite type over a field (not necessarily smooth), there is an analogous spectral sequence from motivic homology to G-theory (the K-theory of coherent sheaves, rather than vector bundles).

Relation to Milnor K-theory

Motivic cohomology provides a rich invariant already for fields. (Note that a field k determines a scheme Spec(k), for which motivic cohomology is defined.) Although motivic cohomology Hⁱ(k, Z(j)) for fields k is far from understood in general, there is a description when i = j:

K_{j}^{M}(k)\cong H^{j}(k,\mathbf {Z} (j)),

where K_j^M(k) is the jth Milnor K-group of k.^[6] Since Milnor K-theory of a field is defined explicitly by generators and relations, this is a useful description of one piece of the motivic cohomology of k.

Map to étale cohomology

Let X be a smooth scheme over a field k, and let m be a positive integer which is invertible in k. Then there is a natural homomorphism (the cycle map) from motivic cohomology to étale cohomology:

H^{i}(X,\mathbf {Z} /m(j))\rightarrow H_{et}^{i}(X,\mathbf {Z} /m(j)),

where Z/m(j) on the right means the étale sheaf (μ_m)^⊗j, with μ_m being the mth roots of unity. This generalizes the cycle map from the Chow ring of a smooth variety to étale cohomology.

A frequent goal in algebraic geometry or number theory is to compute motivic cohomology, whereas étale cohomology is often easier to understand. For example, if the base field k is the complex numbers, then étale cohomology coincides with singular cohomology (with finite coefficients). A powerful result proved by Voevodsky, known as the Beilinson-Lichtenbaum conjecture, says that many motivic cohomology groups are in fact isomorphic to étale cohomology groups. This is a consequence of the norm residue isomorphism theorem. Namely, the Beilinson-Lichtenbaum conjecture (Voevodsky's theorem) says that for a smooth scheme X over a field k and m a positive integer invertible in k, the cycle map

H^{i}(X,\mathbf {Z} /m(j))\rightarrow H_{et}^{i}(X,\mathbf {Z} /m(j))

is an isomorphism for all j ≥ i and is injective for all j ≥ i − 1.^[7]

Relation to motives

For any field k and commutative ring R, Voevodsky defined an R-linear triangulated category called the derived category of motives over k with coefficients in R, DM(k; R). Each scheme X over k determines two objects in DM called the motive of X, M(X), and the compactly supported motive of X, M^c(X); the two are isomorphic if X is proper over k.

One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category. To describe this, first note that there are Tate motivesR(j) in DM(k; R) for all integers j, such that the motive of projective space is a direct sum of Tate motives:

M(\mathbf {P} _{k}^{n})\cong \oplus _{j=0}^{n}R(j)[2j],

where M ↦ M[1] denotes the shift or "translation functor" in the triangulated category DM(k; R). In these terms, motivic cohomology (for example) is given by

H^{i}(X,R(j))\cong {\text{Hom}}_{DM(k;R)}(M(X),R(j)[i])

for every scheme X of finite type over k.

When the coefficients R are the rational numbers, a modern version of a conjecture by Beilinson predicts that the subcategory of compact objects in DM(k; Q) is equivalent to the bounded derived category of an abelian category MM(k), the category of mixed motives over k. In particular, the conjecture would imply that motivic cohomology groups can be identified with Ext groups in the category of mixed motives.^[8] This is far from known. Concretely, Beilinson's conjecture would imply the Beilinson-Soulé conjecture that Hⁱ(X,Q(j)) is zero for i < 0, which is known only in a few cases.

Conversely, a variant of the Beilinson-Soulé conjecture, together with Grothendieck's standard conjectures and Murre's conjectures on Chow motives, would imply the existence of an abelian category MM(k) as the heart of a t-structure on DM(k; Q).^[9] More would be needed in order to identify Ext groups in MM(k) with motivic cohomology.

For k a subfield of the complex numbers, a candidate for the abelian category of mixed motives has been defined by Nori.^[10] If a category MM(k) with the expected properties exists (notably that the Betti realization functor from MM(k) to Q-vector spaces is faithful), then it must be equivalent to Nori's category.

Applications to Arithmetic Geometry

Values of L-functions

Let X be a smooth projective variety over a number field. The Bloch-Kato conjecture on values of L-functions predicts that the order of vanishing of an L-function of X at an integer point is equal to the rank of a suitable motivic cohomology group. This is one of the central problems of number theory, incorporating earlier conjectures by Deligne and Beilinson. The Birch–Swinnerton-Dyer conjecture is a special case. More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of regulators and a height pairing on motivic cohomology.

History

The first clear sign of a possible generalization from Chow groups to a more general motivic cohomology theory for algebraic varieties was Quillen's definition and development of algebraic K-theory (1973), generalizing the Grothendieck group K₀ of vector bundles. In the early 1980s, Beilinson and Soulé observed that Adams operations gave a splitting of algebraic K-theory tensored with the rationals; the summands are now called motivic cohomology (with rational coefficients). Beilinson and Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology. Most but not all of their conjectures have now been proved.

Bloch's definition of higher Chow groups (1986) was the first integral (as opposed to rational) definition of motivic homology for schemes over a field k (and hence motivic cohomology, in the case of smooth schemes). The definition of higher Chow groups of X is a natural generalization of the definition of Chow groups, involving algebraic cycles on the product of X with affine space which meet a set of hyperplanes (viewed as the faces of a simplex) in the expected dimension.

Finally, Voevodsky (building on his work with Suslin) defined the four types of motivic homology and motivic cohomology in 2000, along with the derived category of motives. Related categories were also defined by Hanamura and Levine.

The work of Elmanto and Morrow ^[11] has extended the construction of motivic cohomology to arbitrary quasi-compact, quasi-separated schemes over a field.

Notes

↑ Bloch, Algebraic cycles and higher K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and Proposition 4.2.9.
↑ Voevodsky, Triangulated categories of motives over a field, section 2.2.
↑ Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Example 13.11.
↑ Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Theorem 4.1.
↑ Levine, K-theory and motivic cohomology of schemes I, eq. (2.9) and Theorem 14.7.
↑ Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Theorem 5.1.
↑ Voevodsky, On motivic cohomology with Z/l coefficients, Theorem 6.17.
↑ Jannsen, Motivic sheaves and filtrations on Chow groups, Conjecture 4.1.
↑ Hanamura, Mixed motives and algebraic cycles III, Theorem 3.4.
↑ Nori, Lectures at TIFR; Huber and Müller-Stach, On the relation between Nori motives and Kontsevich periods.
↑ Elmanto, Elden; Morrow, Matthew (2023). "Motivic cohomology of equicharacteristic schemes". arXiv: 2309.08463 .{{cite journal}}: Cite journal requires |journal= (help)

Related Research Articles

<span class="mw-page-title-main">Vladimir Voevodsky</span>

Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.

In mathematics, the Milnor conjecture was a proposal by John Milnor (1970) of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky.

In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil. Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category.

In algebraic geometry, the Chow groups of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.

In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.

In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968). The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.

In algebraic geometry and algebraic topology, branches of mathematics, $A 1$ homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval $[0, 1]$ , which is not an algebraic variety, with the affine line $A 1$ , which is. The theory has seen spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.

In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p. The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.

In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime $and any natural number . John Milnor speculated that this theorem might be true for and all, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch-Kato conjecture or the motivic Bloch-Kato conjecture to distinguish it from the Bloch-Kato conjecture on values of L -functions . The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.$

In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.

Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.

In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences to the category of abelian groups.

References

Bloch, Spencer (1986), "Algebraic cycles and higher K-theory", Advances in Mathematics , 61 (3): 267–304, doi: 10.1016/0001-8708(86)90081-2 , ISSN 0001-8708, MR 0852815
Hanamura, Masaki (1999), "Mixed motives and algebraic cycles III", Mathematical Research Letters, 6: 61–82, doi: 10.4310/MRL.1999.v6.n1.a5 , MR 1682709
Jannsen, Uwe (1994), "Motivic sheaves and filtrations on Chow groups", Motives, Providence, R.I.: American Mathematical Society, pp. 245–302, ISBN 978-0-8218-1637-0, MR 1265533
Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
Voevodsky, Vladimir (2000), "Triangulated categories of motives over a field", Cycles, Transfers, and Motivic Homology Theories, Princeton University Press, pp. 188–238, ISBN 9781400837120, MR 1764202
Voevodsky, Vladimir (2011), "On motivic cohomology with Z/l coefficients", Annals of Mathematics, 174: 401–438, arXiv: 0805.4430 , doi:10.4007/annals.2011.174.1.11, MR 2811603, S2CID 15583705
Levine, Marc (July 12, 2022). "WATCH: Motivic Cohomology: past, present and future" (video). youtube.com. International Mathematical Union.

External links

Huber, Annette; Müller-Stach, Stefan (2011), On the relation between Nori motives and Kontsevich periods, arXiv: 1105.0865 , Bibcode:2011arXiv1105.0865H
Levine, Marc, K-theory and motivic cohomology of schemes I (PDF)
Nori, Madhav, Lectures at TIFR, archived from the original on 22 Sep 2016
Harrer Daniel, Comparison of the Categories of Motives defined by Voevodsky and Nori
Wiesława Nizioł, p-adic motivic cohomology in arithmetic

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Bloch, Algebraic cycles and higher K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and Proposition 4.2.9.

[2] Voevodsky, Triangulated categories of motives over a field, section 2.2.

[3] Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Example 13.11.

[4] Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Theorem 4.1.

[5] Levine, K-theory and motivic cohomology of schemes I, eq. (2.9) and Theorem 14.7.

[6] Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Theorem 5.1.

[7] Voevodsky, On motivic cohomology with Z/l coefficients, Theorem 6.17.

[8] Jannsen, Motivic sheaves and filtrations on Chow groups, Conjecture 4.1.

[9] Hanamura, Mixed motives and algebraic cycles III, Theorem 3.4.

[10] Nori, Lectures at TIFR; Huber and Müller-Stach, On the relation between Nori motives and Kontsevich periods.

[11] Elmanto, Elden; Morrow, Matthew (2023). "Motivic cohomology of equicharacteristic schemes". arXiv: 2309.08463 .{{cite journal}}: Cite journal requires |journal= (help)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]