Advanced Studies in Pure Mathematics

Remarks on the Milnor conjecture over schemes

Asher Auel

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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 $\neq 2$). 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.

Article information

Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 1-30

Received: 17 May 2011
Revised: 15 October 2011
First available in Project Euclid: 24 October 2018

Permanent link to this document euclid.aspm/1540417812

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11-02: Research exposition (monographs, survey articles) 19-02: Research exposition (monographs, survey articles)
Secondary: 11E04: Quadratic forms over general fields 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] 11E88: Quadratic spaces; Clifford algebras [See also 15A63, 15A66] 14F22: Brauer groups of schemes [See also 12G05, 16K50] 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] 16K50: Brauer groups [See also 12G05, 14F22] 19G12: Witt groups of rings [See also 11E81] 19D45: Higher symbols, Milnor $K$-theory

Milnor conjecture quadratic forms Milnor $K$-theory Galois cohomology unramified cohomology cohomological invariants Brauer group


Auel, Asher. Remarks on the Milnor conjecture over schemes. Galois–Teichmüller Theory and Arithmetic Geometry, 1--30, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310001.

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