Norm residue isomorphism theorem

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 ${\displaystyle \ell }$ and any natural number ${\displaystyle n}$. John Milnor^[1] speculated that this theorem might be true for ${\displaystyle \ell =2}$ and all ${\displaystyle n}$, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato^[2] 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.^[3] The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.

For any integer ℓ invertible in a field ${\displaystyle k}$ there is a map ${\displaystyle \partial :k^{\times }\rightarrow H^{1}(k,\mu _{\ell })}$ where ${\displaystyle \mu _{\ell }}$ denotes the Galois module of ℓ-th roots of unity in some separable closure of k. It induces an isomorphism ${\displaystyle k^{\times }/(k^{\times })^{\ell }\cong H^{1}(k,\mu _{\ell })}$. The first hint that this is related to K-theory is that ${\displaystyle k^{\times }}$ is the group K₁(k). Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map ${\displaystyle \partial }$ to maps:

${\displaystyle \partial ^{n}:k^{\times }\otimes \cdots \otimes k^{\times }\rightarrow H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n}).}$

These maps have the property that, for every element a in ${\displaystyle k\setminus \{0,1\}}$, ${\displaystyle \partial ^{n}(\ldots ,a,\ldots ,1-a,\ldots )}$ vanishes. This is the defining relation of Milnor K-theory. Specifically, Milnor K-theory is defined to be the graded parts of the ring:

${\displaystyle K_{*}^{M}(k)=T(k^{\times })/(\{a\otimes (1-a)\colon a\in k\setminus \{0,1\}\}),}$

where ${\displaystyle T(k^{\times })}$ is the tensor algebra of the multiplicative group${\displaystyle k^{\times }}$ and the quotient is by the two-sided ideal generated by all elements of the form ${\displaystyle a\otimes (1-a)}$. Therefore the map ${\displaystyle \partial ^{n}}$ factors through a map:

${\displaystyle \partial ^{n}\colon K_{n}^{M}(k)\to H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n}).}$

This map is called the Galois symbol or norm residue map.^[4][5][6] Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through ${\displaystyle K_{n}^{M}(k)/\ell }$.

The norm residue isomorphism theorem (or Bloch–Kato conjecture) states that for a field k and an integer ℓ that is invertible in k, the norm residue map

${\displaystyle \partial ^{n}:K_{n}^{M}(k)/\ell \to H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n})}$

from Milnor K-theory mod-ℓ to étale cohomology is an isomorphism. The case ℓ = 2 is the Milnor conjecture, and the case n = 2 is the Merkurjev–Suslin theorem.^[6][7]

The étale cohomology of a field is identical to Galois cohomology, so the conjecture equates the ℓth cotorsion (the quotient by the subgroup of ℓ-divisible elements) of the Milnor K-group of a fieldk with the Galois cohomology of k with coefficients in the Galois module of ℓth roots of unity. The point of the conjecture is that there are properties that are easily seen for Milnor K-groups but not for Galois cohomology, and vice versa; the norm residue isomorphism theorem makes it possible to apply techniques applicable to the object on one side of the isomorphism to the object on the other side of the isomorphism.

The case when n is 0 is trivial, and the case when n = 1 follows easily from Hilbert's Theorem 90. The case n = 2 and ℓ = 2 was proved by (Merkurjev 1981) harv error: no target: CITEREFMerkurjev1981 (help). An important advance was the case n = 2 and ℓ arbitrary. This case was proved by (Merkurjev & Suslin 1982) harv error: no target: CITEREFMerkurjevSuslin1982 (help) and is known as the Merkurjev–Suslin theorem. Later, Merkurjev and Suslin, and independently, Rost, proved the case n = 3 and ℓ = 2(Merkurjev & Suslin 1991) harv error: no target: CITEREFMerkurjevSuslin1991 (help)(Rost 1986) harv error: no target: CITEREFRost1986 (help).

The name "norm residue" originally referred to the Hilbert symbol${\displaystyle (a_{1},a_{2})}$, which takes values in the Brauer group of k (when the field contains all ℓ-th roots of unity). Its usage here is in analogy with standard local class field theory and is expected to be part of an (as yet undeveloped) "higher" class field theory.

The norm residue isomorphism theorem implies the Quillen–Lichtenbaum conjecture.

Let X be a smooth variety over a field containing ${\displaystyle 1/\ell }$. Beilinson and Lichtenbaum conjectured that the motivic cohomology group ${\displaystyle H^{p,q}(X,\mathbf {Z} /\ell )}$ is isomorphic to the étale cohomology group ${\displaystyle H_{\rm {{\acute {e}}t}}^{p}(X,\mu _{\ell }^{\otimes q})}$ when p≤q. This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.

• Bloch, Spencer; Kato, Kazuya (1986). "p-adic etale cohomology". Publications Mathématiques de l'IHÉS. 63: 107–152. doi:10.1007/bf02831624.
• Borghesi, Simone (2000), Algebraic Morava K-theories and the higher degree formula, Preprint
• Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. Vol. 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002.
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
• Milnor, John (1970). "Algebraic K-theory and quadratic forms". Inventiones Mathematicae. 9 (4): 318–344. Bibcode:1970InMat...9..318M. doi:10.1007/bf01425486.
• Rost, Markus (1998). "Chain lemma for splitting fields of symbols".
• Srinivas, V. (2008). Algebraic K-theory. Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.). Boston, MA: Birkhäuser. ISBN 978-0-8176-4736-0. Zbl 1125.19300.
• Voevodsky, Vladimir (1995), Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories, Preprint, CiteSeerX 10.1.1.154.922
• Voevodsky, Vladimir (1996), The Milnor Conjecture, Preprint
• Voevodsky, Vladimir (2001), On 2-torsion in motivic cohomology, Preprint, arXiv:math/0107110, Bibcode:2001math......7110V
• Voevodsky, Vladimir (2003a), "Reduced power operations in motivic cohomology", Institut des Hautes Études Scientifiques. Publications Mathématiques, 98 (1): 1–57, arXiv:math/0107109, doi:10.1007/s10240-003-0009-z, ISSN 0073-8301, MR 2031198
• Voevodsky, Vladimir (2003b), "Motivic cohomology with Z/2-coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques, 98 (1): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301, MR 2031199
• Voevodsky, Vladimir (2008). "On motivic cohomology with Z/L coefficients". arXiv:0805.4430 [math.AG].
• Weibel, Charles (2009). "The norm residue isomorphism theorem". Journal of Topology. 2 (2): 346–372. doi:10.1112/jtopol/jtp013. MR 2529300.
• Voevodsky, Vladimir (2011). "On motivic cohomology with Z/L-coefficients". Annals of Mathematics. 174 (1): 401–438. arXiv:0805.4430. doi:10.4007/annals.2011.174.1.11.