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QUOTIENTS OF ABSOLUTE GALOIS GROUPS WHICH DETERMINE THE ENTIRE GALOIS COHOMOLOGY
"... Abstract. For prime power q = p d and a field F containing a root of unity of order q we show that the Galois cohomology ring H ∗ (GF, Z/q) is determined by a quotient G [3] F of the absolute Galois group GF related to its descending qcentral sequence. Conversely, we show that G [3] F is determined ..."
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Abstract. For prime power q = p d and a field F containing a root of unity of order q we show that the Galois cohomology ring H ∗ (GF, Z/q) is determined by a quotient G [3] F of the absolute Galois group GF related to its descending qcentral sequence. Conversely, we show that G [3] F is determined by the lower cohomology of GF. This is used to give new examples of prop groups which do not occur as absolute Galois groups of fields. 1.
2004, ‘The local structure of algebraic Ktheory
"... Algebraic Ktheory draws its importance from its effective codification of a mathematical phenomenon which occurs in as separate parts of mathematics as number theory, geometric topology, operator algebra, homotopy theory and algebraic geometry. In reductionistic language the phenomenon can be phras ..."
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Algebraic Ktheory draws its importance from its effective codification of a mathematical phenomenon which occurs in as separate parts of mathematics as number theory, geometric topology, operator algebra, homotopy theory and algebraic geometry. In reductionistic language the phenomenon can be phrased as there is no canonical choice of coordinates. As such, it is a metatheme for mathematics, but the successful codification of this phenomenon in homotopytheoretic terms is what has made algebraic Ktheory into a valuable part of mathematics. For a further discussion of algebraic Ktheory we refer the reader to chapter I below. Calculations of algebraic Ktheory are very rare, and hard to get by. So any device that allows you to get new results is exciting. These notes describe one way to get such results. Assume for the moment that we know what algebraic Ktheory is, how does it vary with its input? The idea is that algebraic Ktheory is like an analytic function, and we have this other analytic function called topological cyclic homology (T C) invented by Bökstedt, Hsiang and
On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results
"... Abstract. Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the padic group ring Zp[G], we prove many new cases of the ppart of the equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Sp ..."
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Abstract. Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the padic group ring Zp[G], we prove many new cases of the ppart of the equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Spec(L))(r),Z[G]). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic Kgroups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic Kgroups of the ring of integers in L. 1.
Introduction to birational anabelian geometry, Current Developments in Algebraic Geometry
 MSRI publications
"... Abstract. We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function ..."
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Abstract. We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function
NORM VARIETIES AND THE CHAIN LEMMA (AFTER MARKUS ROST)
"... The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 0.1 and the Norm Principle 0.3. These are the steps needed to complete the published verification of the BlochKato conjecture, that the norm residue maps are isomorphisms KM n (k)/p ≃ → Hn et(k, Z/p) for eve ..."
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The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 0.1 and the Norm Principle 0.3. These are the steps needed to complete the published verification of the BlochKato conjecture, that the norm residue maps are isomorphisms KM n (k)/p ≃ → Hn et(k, Z/p) for every prime p, every n and every field k containing 1/p. Throughout this paper, p is a fixed prime, and k is a field of characteristic 0, containing the pth roots of unity. We fix an integer n ≥ 2 and an ntuple (a1,..., an) of units in k, such that the symbol {a} = {a1,..., an} is nontrivial in the Milnor Kgroup KM n (k)/p. Associated to this data are several notions. A field F over k is a splitting field for {a} if {a}F = 0 in KM n (F)/p. A variety X over k is called a splitting variety if its function field is a splitting field; X is pgeneric if any splitting field F has a finite extension E/F of degree prime to p with X(E) ̸ = ∅. A norm variety for {a} is a smooth projective pgeneric splitting variety for {a} of dimension pn−1−1. The following sequence of theorems reduces the BlochKato conjecture to the
Periodicity of hermitian Kgroups
 In preparation
"... 0. Introduction and statements of main results By the fundamental work of Bott [10] it is known that the homotopy groups of classical Lie groups are periodic, of period 2 or 8. For instance, the general linear and symplectic groups satisfy the isomorphisms: πn(GL(R)) ∼ = πn+8(GL(R)) ..."
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0. Introduction and statements of main results By the fundamental work of Bott [10] it is known that the homotopy groups of classical Lie groups are periodic, of period 2 or 8. For instance, the general linear and symplectic groups satisfy the isomorphisms: πn(GL(R)) ∼ = πn+8(GL(R))
Motivic Homotopy Theory
 MILAN JOURNAL OF MATHEMATICS
, 2008
"... We give an informal discussion of the roots and accomplishments of motivic homotopy theory. ..."
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We give an informal discussion of the roots and accomplishments of motivic homotopy theory.