Surjectivity of the total Clifford invariant and Brauer dimension

Auel, Asher

doi:10.1016/j.jalgebra.2015.06.043

Cited by 5 publications

(2 citation statements)

References 62 publications

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“…Their existence was ultimately proved in [OVV07]. Unlike in the case of Stiefel-Whitney classes, it does not seem to be clear how these isomorphisms may be generalized to varieties (c. f. [Aue11a]).…”

Section: Stiefel-whitney Classes Over Fieldsmentioning

confidence: 99%

Witt groups of curves and surfaces

Zibrowius

2014

Math. Z.

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We study Witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two. In both dimensions, we determine both the classical Witt group and Balmer's shifted Witt groups. In the case of curves, the results are supplemented with a complete description of the (shifted) Grothendieck-Witt groups.In a second step, we analyse the relationship of Witt groups of smooth complex curves and surfaces with their real topological K-groups. They turn out to be surprisingly close: for all curves and for all projective surfaces of geometric genus zero, the Witt groups may be identified with the quotients of their even KO-groups by the images of their complex topological K-groups under realification.

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Section: Stiefel-whitney Classes Over Fieldsmentioning

confidence: 99%

Witt groups of curves and surfaces

Zibrowius

2014

Math. Z.

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“…Finally, as a consequence of [8,Cor. 3.4], the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer over any scheme X satisfying: 2 Br(X) is generated by quaternion Azumaya algebras (i.e.…”

Section: Positive Resultsmentioning

confidence: 98%

Remarks on the Milnor conjecture over schemes

Auel

Advanced Studies in Pure Mathematics

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The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic = 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of the unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.

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The topological period-index problem over 6-complexes

Antieau

Williams

2013

Journal of Topology

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By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

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Surjectivity of the total Clifford invariant and Brauer dimension

Cited by 5 publications

References 62 publications

Witt groups of curves and surfaces

Witt groups of curves and surfaces

Remarks on the Milnor conjecture over schemes

The topological period-index problem over 6-complexes

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