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Lecture notes on motivic cohomology
 of Clay Mathematics Monographs. American Mathematical Society
, 2006
"... From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by ..."
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From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by
Volumes of hyperbolic manifolds and mixed Tate motives
 J. Amer. Math. Soc
, 1999
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Unramified cohomology of quadrics, I
, 1997
"... Given a quadric X over a field F of characteristic ̸ = 2, we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 Galois cohomology of F to the unramified mod 2 cohomology of F (X), when dim X> 10 and in several smallerdimensional cases. Applications of these results t ..."
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Cited by 23 (3 self)
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Given a quadric X over a field F of characteristic ̸ = 2, we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 Galois cohomology of F to the unramified mod 2 cohomology of F (X), when dim X> 10 and in several smallerdimensional cases. Applications of these results to
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.
Galois module structure of Galois cohomology and partial EulerPoincaré characteristics
, 2006
"... Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F. Using the BlochKato Conjecture we determine the structure of the cohomology group H n (U, Fp) as an Fp[GF /U]module for all n ∈ N. Previously this stru ..."
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Cited by 18 (13 self)
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Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F. Using the BlochKato Conjecture we determine the structure of the cohomology group H n (U, Fp) as an Fp[GF /U]module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H 1 (U, Fp) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. We apply these results to study partial EulerPoincaré characteristics of open subgroups N of the maximal prop quotient T of GF. We extend the notion of a partial EulerPoincaré characteristic to this case and we show that the nth partial EulerPoincaré characteristic Θn(N) is determined only by Θn(T) and the conorm in H n (T, Fp).
Suite spectral d'Adams et invariants cohomologiques des formes quadratiques
 C.R. Acad. Sci. Ser
, 1999
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On the cohomology of Galois groups determined by Witt rings
 Adv. Math
, 1999
"... Abstract. Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group GF (called the Wgroup of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H ∗ (GF, F2) contains t ..."
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Cited by 15 (10 self)
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Abstract. Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group GF (called the Wgroup of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H ∗ (GF, F2) contains the mod 2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of GF reduces to calculating the equivariant cohomology H ∗ E(XF,F2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal Wgroups. 1.
Algebras of odd degree with involution, trace forms and dihedral extensions
 ISRAEL J. MATH
, 1996
"... A 3fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Su ..."
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Cited by 14 (2 self)
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A 3fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.