In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the norm residue isomorphism theorem is a long-sought result relating
Milnor ''K''-theory and
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
. 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 form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s,
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
and the theory of
motives. The theorem asserts that a certain statement holds true for any prime
and any natural number
.
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
[Milnor (1970)] speculated that this theorem might be true for
and all
, and this question became known as
Milnor's conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wit ...
. The general case was conjectured by
Spencer Bloch
Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Departm ...
and
Kazuya Kato 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.
[Bloch, Spencer and Kato, Kazuya, "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, 333–400, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990.] The norm residue isomorphism theorem was proved by
Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (, ; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal ...
using a number of highly innovative results of
Markus Rost.
Statement
For any integer â„“ invertible in a field
there is a map
where
denotes the
Galois module of â„“-th
roots of unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...
in some
separable closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
of ''k''. It induces an isomorphism
. The first hint that this is related to ''K''-theory is that
is the group ''K''
1(''k''). Taking the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
s and applying the multiplicativity of
étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
yields an extension of the map
to maps:
:
These maps have the property that, for every element ''a'' in
,
vanishes. This is the defining relation of Milnor ''K''-theory. Specifically, Milnor ''K''-theory is defined to be the graded parts of the ring:
:
where
is the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
of the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
and the quotient is by the
two-sided ideal
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even n ...
generated by all elements of the form
. Therefore the map
factors through a map:
:
This map is called the Galois symbol or norm residue map.
[Srinivas (1996) p.146][Gille & Szamuely (2006) p.108][Efrat (2006) p.221] Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through
.
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
:
from
Milnor K-theory In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic ...
mod-â„“ to
étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
is an isomorphism. The case is the
Milnor conjecture, and the case is the Merkurjev–Suslin theorem.
[Srinivas (1996) pp.145-193]
History
The étale cohomology of a field is identical to Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
, so the conjecture equates the â„“th cotorsion (the quotient by the subgroup of â„“-divisible elements) of the Milnor ''K''-group of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k'' with the Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
of ''k'' with coefficients in the Galois module of â„“th 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 follows easily from Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of field (mathematics), fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' ...
. The case and was proved by . An important advance was the case and ℓ arbitrary. This case was proved by and is known as the Merkurjev–Suslin theorem. Later, Merkurjev and Suslin, and independently, Rost, proved the case and .
The name "norm residue" originally referred to the Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of real number, reals or p-adic numbers. It is related to rec ...
, which takes values in the Brauer group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
of ''k'' (when the field contains all â„“-th roots of unity). Its usage here is in analogy with standard local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
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
In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for ...
.
History of the proof
Milnor's conjecture was proved by Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (, ; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal ...
.[Voevodsky, Vladimir, "Motivic cohomology with Z/2-coefficients", ''Publ. Math. Inst. Hautes Études Sci.'' No. 98 (2003), 59–104.]
Later Voevodsky proved the general Bloch–Kato conjecture.[Voevodsky (2010)]
The starting point for the proof is a series of conjectures due to and . They conjectured the existence of ''motivic complexes'', complexes of sheaves whose cohomology was related to motivic cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geome ...
. Among the conjectural properties of these complexes were three properties: one connecting their Zariski cohomology to Milnor's K-theory, one connecting their etale cohomology to cohomology with coefficients in the sheaves of roots of unity and one connecting their Zariski cohomology to their etale cohomology. These three properties implied, as a very special case, that the norm residue map should be an isomorphism. The essential characteristic of the proof is that it uses the induction on the "weight" (which equals the dimension of the cohomology group in the conjecture) where the inductive step requires knowing not only the statement of Bloch-Kato conjecture but the much more general statement that contains a large part of the Beilinson-Lichtenbaum conjectures. It often occurs in proofs by induction that the statement being proved has to be strengthened in order to prove the inductive step. In this case the strengthening that was needed required the development of a very large amount of new mathematics.
The earliest proof of Milnor's conjecture is contained in a 1995 preprint of Voevodsky and is inspired by the idea that there should be algebraic analogs of Morava ''K''-theory (these algebraic Morava K-theories
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is s ...
were later constructed by Simone Borghesi[Borghesi (2000)]). In a 1996 preprint, Voevodsky was able to remove Morava ''K''-theory from the picture by introducing instead algebraic cobordisms and using some of their properties that were not proved at that time (these properties were proved later). The constructions of 1995 and 1996 preprints are now known to be correct but the first completed proof of Milnor's conjecture used a somewhat different scheme.
It is also the scheme that the proof of the full Bloch–Kato conjecture follows. It was devised by Voevodsky a few months after the 1996 preprint appeared. Implementing this scheme required making substantial advances in the field of motivic homotopy theory
In music, a motif () or motive is a short musical idea, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition. The motif is the smallest structural unit ...
as well as finding a way to build algebraic varieties with a specified list of properties. From the motivic homotopy theory the proof required the following:
#A construction of the motivic analog of the basic ingredient of the Spanier–Whitehead duality In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space ''X'' may be considered as dual to its complement in the ''n''-sphere, where ''n'' is large enough. Its ori ...
in the form of the motivic fundamental class as a morphism from the motivic sphere to the Thom space of the motivic normal bundle over a smooth projective algebraic variety.
#A construction of the motivic analog of the Steenrod algebra
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology.
Life
He was born in Dayton, Ohio, and educated at Miami University and University of ...
.
#A proof of the proposition stating that over a field of characteristic zero the motivic Steenrod algebra
In music, a motif () or motive is a short musical idea, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition. The motif is the smallest structural unit ...
characterizes all bi-stable cohomology operations in the motivic cohomology.
The first two constructions were developed by Voevodsky by 2003. Combined with the results that had been known since late 1980s, they were sufficient to reprove the Milnor conjecture.
Also in 2003, Voevodsky published on the web a preprint that nearly contained a proof of the general theorem. It followed the original scheme but was missing the proofs of three statements. Two of these statements were related to the properties of the motivic Steenrod operations and required the third fact above, while the third one required then-unknown facts about "norm varieties". The properties that these varieties were required to have had been formulated by Voevodsky in 1997, and the varieties themselves had been constructed by Markus Rost in 1998–2003. The proof that they have the required properties was completed by Andrei Suslin
Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University.
He ...
and Seva Joukhovitski in 2006.
The third fact above required the development of new techniques in motivic homotopy theory. The goal was to prove that a functor, which was not assumed to commute with limits or colimits, preserved weak equivalences between objects of a certain form. One of the main difficulties there was that the standard approach to the study of weak equivalences is based on Bousfield–Quillen factorization systems and model category
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
structures, and these were inadequate. Other methods had to be developed, and this work was completed by Voevodsky only in 2008.
In the course of developing these techniques, it became clear that the first statement used without proof in Voevodsky's 2003 preprint is false. The proof had to be modified slightly to accommodate the corrected form of that statement. While Voevodsky continued to work out the final details of the proofs of the main theorems about motivic Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
s, Charles Weibel
Charles Alexander Weibel (born October 28, 1950, in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
Weibel studied physics and mathematics at the University of Michiga ...
invented an approach to correct the place in the proof that had to modified. Weibel also published in 2009 a paper that contained a summary of Voevodsky's constructions combined with the correction that he discovered.
Beilinson–Lichtenbaum conjecture
Let ''X'' be a smooth variety over a field containing . Beilinson and Lichtenbaum conjectured that the motivic cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geome ...
group is isomorphic to the étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
group when ''p''≤''q''. This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.
References
Bibliography
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* {{cite journal, last1=Voevodsky, first1=Vladimir, title=On motivic cohomology with {{not a typo, Z/L-coefficients , journal=Annals of Mathematics, date=2011, volume=174, issue=1, pages=401–438, doi=10.4007/annals.2011.174.1.11, url=http://annals.math.princeton.edu/2011/174-1/p11, arxiv=0805.4430
Conjectures that have been proved
Algebraic K-theory
Theorems in algebraic topology
Theorems in algebra