Fall 2006 Lectures on the proof of the Bloch-Kato Conjecture

C. Weibel

The Norm Residue Theorem asserts that the following is true: For an odd prime l, and a field k containing 1/l,

1) the Milnor K-theory KMn(k)/l is isomorphic to the étale cohomology Hn(k,μln)
of the field k with coefficients in the twists of μl.

2) For n ≤ i, the motivic cohomology group Hn,i(X,Z/l) is isomorphic to the etale cohomology group Hn(X,μli).

The case l=2 is a theorem of Voevodsky, established in [MC/2].
The case n=1 is a well known part of Kummer theory. The cases n=2 (and n=3, l=2) were established by 1990.
The equivalence of the two assertions is due to Suslin-Voevodsky in characteristic 0, and is also true in characteristic p>0.

Status as of October 2006: The 2003 preprint [MC/l] sketches a proof, modulo two missing lemmas (2.2, 2.3).
The preprint also assumed that a construction of Rost in [R-CL] satisfied certain properties.
These properties have since been verified in the Suslin-Joukhovitsky paper [SJ].

Status as of January 2009: All the papers involved in the proof, including [MC/l], [V-cancel], [V-over] as well as [HW], [W-axioms] and [W-patch], have now been submitted for publication. Several of these have been refereed and accepted.

Status as of Summer 2010: All of the papers involved in the proof have been accepted for publication.
Haesemeyer and I are writing up the entire proof in book form.

During the Fall semester of 2006, I gave a series of lectures at the Institute for Advanced Study on
the status of the "Voevodsky-Rost Theorem," stated above. In the Spring semester, I gave a proof of this theorem.
Here is an outline of those lectures.

October 26: This was the first lecture. I started by describing the current status of the proof, which is described above. By a reduction in [MC/2], it suffices to start with a nonzero symbol a in KMn(k) and produce a splitting variety X of dimension d=ln-1 such that

(*) the étale cohomology group Hn+1,n(k,Z) injects into the corresponding cohomology group for k(X).

By a known argument, a induces a nonzero element δ in Hn,n-1(χ,Z/l), where χ is a simplicial scheme formed from X. Applying a motivic cohomology operation to δ, we obtain an integral class μ in H2b+1,b(χ,Z), where b=(li-1)/(l-1).

The goal of the November lectures (which was achieved) will be to prove that μ is nonzero when X is a splitting variety.
I also showed how μ defined an element ρ of CHb(X×X). Rost calls ρ the basic correspondence of X, and studies it in [R-BC]. The power ρl-1 is an element of CHd(X× X), i.e., a classical Chow correspondence from X to itself.

Nov.2: I gave the following axiomatic presentation of the motivic operations Pi:

The operation Pi has bidegree (2li-1,li-1) with P0x=x.
Pix=x^l if x has bidegree (2i,i).
Pix=0 if x has bidegree (p,q) and q≤i, p < q+i.
The usual Cartan formula for Pn(xy) and the usual Adem relations hold.
It follows that the Pi and the Bockstein β generate a bigraded ring isomorphic to the topological Steenrod Algebra described in [SE]. In particular, we get derivations Qi in bidegree (2li-1,li-1).

Nov.9: I constructed the cohomology operations Pi on motivic cohomology with coefficients Z/l, l an odd prime.
The presentation was based upon [RPO].
First I observed that, if G is the symmetric group on l letters, the motivic cohomology of BG is H[c,d], where d has bidegree (2l-2,l-1), βc=d and H is the motivic cohomology of k. There is a Künneth formula for the cohomology of X×BG.

Next, I constructed a functorial version of the map from CHe(X) to CHel(X × BG). Roughly speaking, if we replace X by the classifying space Ke of CHe then the image of the canonical element gives an element P of CHel(Ke × BG). By the Künneth formula, P is a polynomial in c,d and we define Pi to be the coefficient of de-i. The coefficient of cde-i is βPi+1.

Nov. 16: Proved that the above construction produces a nonzero integral class μ in H2b+1,b(χ,Z).
This is the black box input used by Rost in his basic correspondences paper [R-BC].
The lecture covered the contents of section 4 (except 4.4) and 6.5-6.7 of [MC/l]. The following construction was used to prove that the degree of sd(X) is divisible by l, and to simplify the proof of 4.1.

For any d-dimensional projective X, there is a classical map Z(d)[2d] → M(X) in DM, or X→Spec k in the category of Chow motives, defined by the cycle (X×Spec k) in CH0(X×Spec k). (It generates the 2d-part of the rational Chow motive of X.) The composition of this map with a zero-cycle Z, thought of as a map M(X)→Z(d)[2d], gives the degree of Z.

Dec. 7: I gave the following set of axioms, and showed that they imply the Bloch-Kato Conjecture.
Axioms: 0. M should be a direct summand of M(X), i.e., a Chow motive. Thus there is a canonical map y: M → M(χ).

  1. The composite M⊗M → X⊗X → χ⊗Ld induces an isomorphism: M* ⊗ Ld = M.
    Here L=Z(1)[2] is the Lefschetz motive. We write y* for the map M(χ)⊗ Lb → M which is dual to y.
  2. The maps y and y* fit into two distinguished triangles:
    D ⊗ Lb → M → M(χ) →
    M(χ)⊗ Lb → M → D →
This material is directly inspired by Rost's paper [R-BC] and essentially covers the material in 6.9-6.15 of [MC/l].
It has since been written up (and published) in the paper [W-axioms]. The first Axiom can be rewritten as:
1. As a Chow motive, the direct summand M=(X,e) is isomorphic to (X,et).

Dec. 14: Following Rost, I constructed a symmetric idempotent e of the Chow motive of X, and defined M to be the Chow motive (X,e). Thus Axioms 0 and 1 are automatically satisfied by M.
I don't know how to verify Axiom 2. This material is due to Rost [R-BC].

We start with the element ρ of CHb(X×X), constructed in the October 26 Lecture from the element μ.
We assume Rost's calculation that c=π*ρl-1 is not divisibly by l, where π is the projection X×X →X and
Theorem: There is an idempotant e in End(X)=CHd(X×X).
It is constructed in the ring CH*(X×X) as a polynomial in ρ and in the π*π*ρj, l < j.
To show this, we considered the special case ρ=H×X-X×H when ce is the sum of the Hi×Hj with i+j=l-1.

March 8, 2007: I introduced the notion of a proper Tate motive, which is a direct sum of Lefschetz motives La[b], b≥0 in the category of motives with coefficients mod l. This category is idempotent complete and (by Cancellation) the Künneth Formula holds for wedges of spaces whose motives are proper Tate motives.

I sketched a proof of Voevodsky's Theorem [V07] that the category of proper Tate motives is closed under symmetric products. A theorem of Suslin-Voevodsky [SV] implies that if K is the space representing H2n,n(-,Z) and n>0, then its motive is the infinite symmetric product of Ln. This proves that the Künneth Formula holds for wedges of copies of K. This is a slight modification of the missing Lemma 2.3 from [MC/l].

April 19, 2007: I finished the proof of the Bloch-Kato conjecture.
This material has since been written up in the preprint [W-patch]

The key new idea is the notion of scalar weight, based upon the action of Z/l on H**(K,Z/l), where K represents some Hp,q(-,Z). When (p,q)=(2n,n) this has an interpretation in terms of infinite symmetic products; terms from SsLn have scalar weight s, and its pure Tate summands La[b] satisfy:

  • a ≥ ns, with equality iff s a ≥ n(l-1) if s=0, with equality iff s=l-1.
    This replaces the missing Lemma 2.2 from [MC/l], and suffices to prove the following replacement of Theorem 2.1 from [MC/l]:

    Theorem: Let φ be a cohomology operation of scalar weight one, from H2n+1,n(X,Z) to H2nl+1,nl(X,Z/l).
    If φ vanishes on suspension elements, then φ is a multiple of βPn.

    The primary application of this is to the cohomology operation constructed in section 3 of [MC/l].
    Now set M=Sl-1A, where A is the fiber of μ:χ → χ(b)[2b+1].
    By construction, φ is nonzero but vanishes on M. The above theorem says that βPn also vanishes on M. This suffices for Voevodsky's proof to go through, showing that the map λ: M(X)→ M is a split surjection, and proving:

  • Theorem (Lemma 5.15 in [MC/l]) M is a summand of M(X) satisfying the above axioms.

    As we saw in the Dec.7 lecture, this establishes the Bloch-Kato conjecture.


    [MC/2] V. Voevodsky, Motivic Cohomology with Z/2 coefficients, Publ. IHES 98 (2003), 1-57.
    [MC/l] V. Voevodsky, On Motivic Cohomology with Z/l coefficients, Annals of Math., to appear. This is a 2009 revision of the 2003 preprint preprint.
    [RPO] V. Voevodsky, Reduced Power operations in Motivic Cohomology, Publ. IHES 98 (2003), 1-57.
    [R-CL] M. Rost, Chain lemma for splitting fields of symbols, preprint, 1998.
    [R-BC] M. Rost, On the Basic Correspondence of a Splitting Variety, in preparation, Fall 2006.
    [SE] N. Steenrod and D. Epstein, Cohomology Operations, Annals of Math Studies 50, 1962.
    [SJ] A. Suslin and S. Joukhovitski, Norm Varieties, JPAA 206 (2006), 245-276.
    [SV] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996) 61-94.
    [V07] V. Voevodsky, Motivic Eilenberg-Mac Lane spaces, Publ. IHES, to appear. This is a revision of the 2007 preprint Operations II.
    References added in 2009:
    [HW] C. Haesemeyer and C. Weibel, Norm Varieties and the chain lemma (after Markus Rost), Proc. Abel Symp. 4 (2009), 95-130.
    [V-cancel] V. Voevodsky, Cancellation theorem, preprint 2002. Documenta Math., to appear.
    [V-over] V. Voevodsky, Motives over simplicial schemes, J. K-theory 5 (2010), 1-38. This 2003 preprint was revised in 2008.
    [W-axioms] C. Weibel, Axioms for the norm residue isomorphism, pp. 427-436 in K-theory and Noncommutative Geometry, European Math. Soc. Pub. House, 2008.
    [W-patch] C. Weibel, The Norm Residue Isomorphism Theorem, J. Topology 2 (2009), 346-372.
    This site maintained by
    Charles Weibel / April 19, 2007 (updated September 2010)