The Algebraic and Geometric Theory of Quadratic Forms

Elman, Richard; Karpenko, Nikita A.; Merkurjev, Alexander

doi:10.1090/coll/056

Cited by 390 publications

(611 citation statements)

References 60 publications

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“…The rank-1 case is trivial, so we can assume that (V, q) has rank at least 2. Since v and w are unimodular, we have v i = 0 = w i for all 1 ≤ i ≤ s. Thus, by an observation of E. Witt (see [EKM08,§II.8…”

Section: Bmentioning

confidence: 84%

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

Scully

2017

Proc. Amer. Math. Soc.

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Abstract. Let R be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over R remain anisotropic after base change to any odd-degree finiteétale extension of R. This generalization of the classical Artin-Springer theorem (concerning the situation where R is a field) was previously established in the case where all residue fields of R are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic = 2 to the case where the ring has at least one residue field which is finite.

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Section: Bmentioning

confidence: 84%

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

Scully

2017

Proc. Amer. Math. Soc.

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“…The preimage of {x} × Γ under the morphism ∆ X × id Y : X × Y → X × X × Y is the reduced scheme {x} × {f (x)}. It follows from [7,Corollary 57.20] …”

Section: Rc-ii the Cycle Module A 0 [Y M] For A Cycle Module M Ovementioning

confidence: 99%

“…Many events in the paper happen in the category of Chow motives (with coefficients in Λ, see [7]). A Chow motive is a pair (X, ρ), where X is a smooth complete variety over F and ρ is a projector (idempotent) in the endomorphism ring of the motive M(X) of X.…”

Section: Introductionmentioning

confidence: 99%

On standard norm varieties

Karpenko

Merkurjev

2013

Ann. Sci. École Norm. Sup.

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Abstract. Let p be a prime integer and F a field of characteristic 0. Let X be the norm variety of a symbol in the Galois cohomology group H n+1 (F, µ ⊗n p ) (for some n ≥ 1), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F (X) has the following property: for any equidimensional variety Y , the change of field homomorphism CH(Y ) → CH(Y F (X) ) of Chow groups with coefficients in integers localized at p is surjective in codimensions < (dim X)/(p−1). One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is A-triviality of X, the property saying that the degree homomorphism on CH 0 (X L ) is injective for any field extension L/F with X(L) = ∅. The proof involves the theory of rational correspondences, due to Markus Rost, reviewed in Appendix.

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“…Now we are going to define our second basic operation. Consider the total cohomological Steenrod operation S • , [2,Chapter XI]. This is a certain endomorphism of the cofunctor Ch of the category of smooth F -varieties to the category of rings.…”

Section: Operations Sq and Stmentioning

confidence: 99%

Hyperbolicity of unitary involutions

Karpenko

2012

Sci. China Math.

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Abstract. We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

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The Algebraic and Geometric Theory of Quadratic Forms

Cited by 390 publications

References 60 publications

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

On standard norm varieties

Hyperbolicity of unitary involutions

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