THE BASIC CORRESPONDENCE OF A SPLITTING VARIETY

MARKUS ROST

Summary of the talk at the 2007 Abel Symposium, Oslo, Norway

We discussed some details of a construction used in the proof of the generalized Milnor conjecture. A general overview had been given by Sasha Merkurjev in the morning. Let me just repeat here the statement of the conjecture:

Consider the norm residue homomorphism

h_{(n,p)} : K_{n}^{M}k/p → H_{et}^{n}(k, μ_{p}^{⊗n})
{a_{1}, …, a_{n}} ↦ (a_{1}) ∪ ⋯ ∪ (a_{n})

from Milnor’s K-groups to Galois cohomology. The generalized Milnor conjecture (aka Milnor-Bloch-Kato conjecture, aka …) states the bijectivity of this map for any prime p, any n, and any field k with char k ≠ p.

In the talk we concentrated on the “basic correspondence” of a splitting variety. The general reference for this is [3]. For norm varieties and their relation to characteristic numbers and cobordism see [2].

In [3] we introduced also the more abstract notion of a “special correspondence” on a variety (with the basic correspondences on norm varieties as only known examples so far). Varieties with special correspondences have been considered further recently in [9].

The following text is an extended version of [4].

The basic correspondence of a splitting variety of a symbol u (see below) is obtained by the following diagram, which is essentially due to Voevodsky:

u ∈ ker[H_{et}^{n}(k, μ_{p}^{⊗(n−1)}) ⟶ H_{et}^{n}(k(X), μ_{p}^{⊗(n−1)})]
≃↑^{j}
H_{M}^{n,n−1}(X , Z/p)
↓β ∘ Q_{1} ∘ ⋯ ∘ Q_{n−2}
μ ∈ H_{M}^{2b+1,b}(X , Z)
↓^{proj}
homology of [ CH^{b}(X) → CH^{b}(X^{2}) → CH^{b}(X^{3})]

Date: August 17, 2007.

Here

u = (a_{1}) ∪ ⋯ ∪ (a_{n}) ∈ H_{et}^{n}(k, μ_{p}^{⊗n})

is a symbol (we assume μ_{p} ⊂ k and fix a generator of μ_{p}) and X is a smooth variety over k over which the symbol is split, i.e., u_{k(X)} = 0.

Furthermore, X is the simplicial scheme

X : X \overleftarrow{←} X^{2} ← ; ← ; ← X^{3} ⋯

The map j relating motivic cohomology of X to Galois cohomology is an isomorphism if one assumes the generalized Milnor conjecture in weight n − 1. For this one uses results from [6].

Then one applies the Milnor operations Q_{i} in motivic cohomology (these can be expressed in terms of the motivic Steenrod operations similarly as in topology) and the Bockstein homomorphism β.

One obtains the class

μ ∈ H_{M}^{2b+1,b}(X , Z), b = \frac{p^{n−1} − 1}{p − 1}

which plays an essential role in Voevodsky’s work on the generalized Milnor conjecture, cf. [7]. If X is a norm variety for the symbol u, Voevodsky uses the class μ to show that X is a generic (up to extensions of degree prime to p) splitting variety for u and to split off from X a certain motive, the so-called generalized Rost motive. (For p = 2 genericity and the construction of the motive can be obtained in a much more elementary way using quadratic forms.) All this is essential for the final proof of the conjecture (involving, as for p = 2, Margolis homology and the so-called “injectivity”, settled in [1], see also [5]).

An important step in handling μ is to verify a certain nontriviality condition. Some ingredients for this part of Voevodsky’s work have not been written up in details yet, but it seems that they will appear soon, cf. [8].

Last year I was able to derive genericity and the construction of the motive from μ in a more ad hoc fashion, cf. [3]. One considers the standard spectral sequence for the simplicial scheme X which leads to the map proj as indicated in the diagram. Then one picks a representative

ρ ∈ CH^{b}(X^{2})

of proj(μ). I call any such element a basic correspondence of the norm variety X of u. Working with ρ, the necessary nontriviality condition reads as

(1)  c(ρ) ≠ 0

where

c(ρ) ∈ Z/pZ

is a certain integer mod p (see [3, Section 5, p. 13] for the definition). I could verify condition (1) “by hand”, so to speak, namely by investigating the specific examples of norm varieties I had constructed earlier in [1].

Once one knows (1), it is surprisingly easy to prove p-genericity of the norm variety using the functoriality of the definition of μ and ρ. The argument, essentially due to Voevodsky, was discussed in the lecture. It is described in [3, Section 6].

The construction of the motive can be done in the general setting of special correspondences, see [3, Section 7]. For p = 2 things become particularly easy, see [3, Section 7.3].

For an illustration, let me describe the basic correspondence in the case n = 2. In this case b = 1. For X we take a Severi-Brauer variety (of dimension p − 1). Thus ρ is an element in the Picard group of X^{2}:

ρ ∈ CH^{1}(X^{2}) = Pic(X^{2})

If we pass to the algebraic closure k̄ of k, then

Xk̄ = Pp−1 ; k̄

and one finds

ρk̄ = π_{0}^{∗}[O(1)] − π_{1}^{∗}[O(1)] mod p Pic(X2 ; k̄)

where

π_{0}, π_{1} : X × X → X

are the projections.

References

[1] M. Rost, Chain lemma for splitting fields of symbols, Preprint, 1998, ⟨www.math.uni-biele feld.de/˜ rost/chain-lemma.html⟩.

[2] , Norm varieties and algebraic cobordism, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 77–85.

[3] , On the basic correspondence of a splitting variety, Preprint, 2006, ⟨www.math.unibielefeld.de/˜ rost/basic-corr.html⟩.

[4] , Norm residue homomorphism, Abstract for the talk at the Mathematische Arbeitstagung, Bonn, June 22-28, 2007, ⟨www.math.uni-bielefeld.de/˜ rost/basic-corr.html⟩.

[5] A. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276.

[6] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189.

[7] V. Voevodsky, Motivic cohomology with Z/l-coefficients, Preprint, 2003, K-theory Preprint Archives, No. 639, ⟨www.math.uiuc.edu/K-theory/0639/⟩.

[8] C. Weibel, Patching the norm residue isomorphism theorem, Preprint, 2007, K-theory Preprint Archives, No. 844, ⟨www.math.uiuc.edu/K-theory/0844/⟩.

[9] K. Zainoulline, Special correspondences and Chow traces of Landweber-Novikov operations, Preprint, 2007, Linear Algebraic Groups and Related Structures, Preprint Server, Bielefeld, ⟨www.math.uni-bielefeld.de/lag/man/267.html⟩.

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

E-mail address: rost at math.uni-bielefeld.de
URL: http://www.math.uni-bielefeld.de/~rost