Introduction

From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors

H^{p,q}(−, A) : Sm/k → Ab

from smooth schemes over a given field k to abelian groups where p and q are integers. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum. The key properties of the motivic cohomology (predicted in [1] and [5]) can be divided into two groups. In the fisrt group there are properties of the motivic cohomology itself - theorems such as the homotopy invariance, Mayer-Vietoris sequence, projective bundle theorem and the Gysin long exact sequence. In the same groups are the yet unproved conjectures such as the Beilinson-Soule vanishing (H^{p,q} = 0 for p < 0) and many others. In the second group are the results relating the motivic cohomology to other known invariants of algebraic varieties and rings. The power of the motivic cohomology as a tool for proving results in algebra and algebraic geometry is lies in the interaction of the results in these two groups - specializing general theorems about motivic cohomology to the cases when they compare to classical invariants one gets new results about these invariants.

The idea of these lectures was to define motivic cohomology and to give careful proofs for the elemntary results in the second group. In this sense they are complimentary to the study of [8] where the emphasis is made on the properties of the motivic cohomology itself. In the process we are forced by the structure of the proofs to deal with the main properties of the motivic cohomology such as homotopy invariance as well. As a result, these lectures cover a considerable portion of the material of [8] but do this from a different view point.

One can distinguish the following main “comparison” results for the motivic cohomology:

1. H^{p,q}(X, A) = 0 for q < 0 and for a connected X one has

H^{p,0}(X, A) = \{ \begin{matrix}A & for p = 0 \\ 0 & for p ≠ 0 \end{matrix}

2. one has

H^{p,1}(X, Z) = \{ \begin{matrix}O^{∗}(X) & for p = 1 \\ P ic(X) & for p = 2 \\ 0 & for p ≠ 1, 2 \end{matrix}

3. for k a field one has H^{p,p}(Spec(k), A) = K_{p}^{M}(k) ⊗ A where K_{p}^{M}(k) is the p-th Milnor K-group of k

4. for S a strictly henselian local scheme over k and n an integer prime to char(k) one has

H^{p,q}(S, Z/n) = \{ \begin{matrix}μ_{n}(S) & for p = 0 \\ 0 & for p ≠ 0 \end{matrix}

5. one has H^{p,q}(X, A) = CH^{q}(X, 2q − p; A) where CH^{i}(X, j; A) are the higher Chow groups of X introduced by S. Bloch in [2], [3]. In particular,

H^{2q,q}(X, A) = CH^{q}(X) ⊗ A

where CH^{q}(X) is the classical Chow group of cycles of codimension q modulo rational equivalence.

This list contains all the comparison results which we would like to refer to as “elementary”. In addition one could include here the Beilinson-Lichtenbaum Conjecture (proved for Z/2-coefficients over fields of characteristic zero) which identifies H^{p,q}(X, Z/l) for p ≤ q with H_{et}^{p}(X, μ_{l}^{⊗q}) and the M. Levine theorem comparing CH^{i}(X, j; Q) with the graded pieces of the gamma filtration in K_{∗}(X) ⊗ Q. Both of these results are out of the scope of these lectures and the interested reader is refered to [6] and [4] respectively.

The lectures of the fall term contain the definition of the motivic cohomology and the proofs for all the comparison results listed above but the last one. The spring term lectures contain some more advanced theory of sheaves with transfers and the proof of the last comparison result.

The definition of the motivic cohomology which we use here goes back to the work of Andrei Suslin in about 1986. As far as I understand when he came up with his idea he was able to prove the first three of our comparison results. In particular the proof of the comparison between the motivic cohomology and Milnor’s K-groups given here is exactly Suslin’s original proof. The proofs of the two other comparison results (4) and (5) are also based on Suslin’s results. In the first case it is his reformulation of the Rigidity Theorem (Theorem ??) and in the second one his moving lemma (to appear in the second part of the lectures).

The main new idea which appeared around 1993 and made it possible to finish the proofs of the more complicated comparisons (4) and (5) is the idea of a sheaf (or presheaf) with transfers originally introduced in [7]. The concept itself is also just a reformulation of some of Suslin’s ideas which he developed in connection with the Rigidity Theorem (Theorem ??). The new development was in the understanding of how one can efficiently do homological algebra in the category of sheaves with transfers.

We need to say a few further words about our approach to the comparison (4). Instead of stating it in the form used above we prove a more detailed result. The motivic cohomology H^{p,q}(X, A) for a given weight q is defined as the hypercohomology (in the Zariski topology) of X with coefficients in a complex of sheaves A(q). This complex is defined not just on the small Zariski site of X (i.e. the category of open subsets of X) but on the big site of all smooth scheme over k with the Zariski topology. In particular, it is a well defined complex of presheaves on the small etale site Et/X of X and it turns out that it is in fact a complex of sheaves on this site. Because of this it makes sense to consider the etale version of the motivic cohomology i.e. the etale hypercohomology of X with coefficients in A(q). We prove (Theorem ??) that the etale motivic cohomology of any X with coefficients in Z/n(q) where n is prime to char(k) are isomorphic to H_{et}^{p}(X, μ_{n}^{⊗q}). Since for a strictly henselian local scheme X the Zariski and the etale motivic cohomology clearly agree this implies in particular the comparison result (4). There is also an analog of (4) for the case of Z/p coefficients where p = char(k) but we do not discuss it here.

This is an expanded and rewritten version of the lecture notes for the lectures on the motivic cohomology given by Vladimir Voevodsky at the Institute for Advanced Study in Princeton in 1999/2000. At the moment only the first half of the notes, corresponding to the fall term of 1999 is available.

References

[1] A. Beilinson, R. MacPherson, and V. Schechtman. Notes on motivic cohomology. Duke Math. J., pages 679–710, 1987.

[2] S. Bloch. Algebraic cycles and higher K-theory. Adv. in Math., 61:267– 304, 1986.

[3] S. Bloch. The moving lemma for higher Chow groups. J. Algebr. Geom., 3(3):537–568, Feb. 1994.

[4] M. Levine. Bloch’s higher Chow groups revisited. Preprint, 1993.

[5] S. Lichtenbaum. Values of zeta-functions at non-negative integers. In Number theory, volume 1068 of Lecture Notes in Math., pages 127–138. Springer-Verlag, 1983.

[6] Vladimir Voevodsky. The Milnor Conjecture. www.math.uiuc.edu/Ktheory/170, 1996.

[7] Vladimir Voevodsky. Cohomological theory of presheaves with transfers. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 87–137. Princeton Univ. Press, 2000.

[8] Vladimir Voevodsky, Eric M. Friedlander, and Andrei Suslin. Cycles, transfers and motivic homology theories. Princeton University Press, 2000.