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

of the field k with coefficients in the twists of μ

2) For * n ≤ i, * the motivic cohomology group
H^{n,i}(X,Z/l) is isomorphic to the etale cohomology group
H^{n}(X,μ_{l}^{i}).

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
K^{M}_{n}(k) and produce a splitting variety X of
dimension d=l^{n}-1 such that

By a known argument, **a** induces a nonzero element δ in
H^{n,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 H^{2b+1,b}(χ,Z), where
b=(l^{i}-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 CH^{b}(X×X).
Rost calls ρ the *basic correspondence* of X, and studies it
in [R-BC]. The power ρ^{l-1} is an element of
CH^{d}(X× X), i.e., a classical Chow correspondence from
X to itself.

**Nov.2:**
I gave the following axiomatic presentation
of the motivic operations P^{i}:

P

P

The usual Cartan formula for P

**Nov.9:** I constructed the cohomology operations P^{i} 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
CH^{e}(X) to CH^{el}(X × BG). Roughly speaking,
if we replace X by the classifying space K_{e} of CH^{e}
then the image of the canonical element gives an element *P* of
CH^{el}(K_{e} × BG). By the Künneth formula,
*P* is a polynomial in c,d and we define P^{i} to be the
coefficient of d^{e-i}. The coefficient of cd^{e-i} is
βP^{i+1}.

**Nov. 16:**
Proved that the above construction produces a nonzero
integral class μ in H^{2b+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
s_{d}(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 CH^{0}(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).

- The composite M⊗M → X⊗X →
χ⊗L
^{d}induces an isomorphism: M* ⊗ L^{d}= M.

Here L=Z(1)[2] is the Lefschetz motive. We write y* for the map M(χ)⊗ L^{b}→ M which is dual to y. - The maps y and y* fit into two distinguished triangles:
D ⊗ L ^{b}→ M → M(χ) →M(χ)⊗ L ^{b}→ M → D →

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,e

**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 CH^{b}(X×X),
constructed in the October 26 Lecture from the element μ.

We assume Rost's calculation that c=π_{*}ρ^{l-1}
is not divisibly by l, where π is the projection X×X →X and

**Theorem: There is an idempotant e in End(X)=CH ^{d}(X×X).**

It is constructed in the ring CH

To show this, we considered the special case ρ=H×X-X×H when ce is the sum of the H

**March 8, 2007:** I introduced the notion of a *proper Tate
motive*, which is a direct sum of Lefschetz motives
L^{a}[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
H^{2n,n}(-,Z) and n>0, then its motive is the infinite symmetric
product of L^{n}. 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 H^{p,q}(-,Z).
When (p,q)=(2n,n) this has an interpretation in terms of infinite symmetic
products; terms from S^{s}L^{n} have scalar weight s,
and its pure Tate summands L^{a}[b] satisfy:

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
H^{2n+1,n}(X,Z) to H^{2nl+1,nl}(X,Z/l).

If φ vanishes on suspension elements, then
φ is a multiple of βP^{n}.

The primary application of this is to the cohomology operation constructed
in section 3 of [MC/l].

Now set M=S^{l-1}A, 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 βP^{n} 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:

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

[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,

[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)