## Advanced Studies in Pure Mathematics

### Remarks on the Milnor conjecture over schemes

Asher Auel

#### Abstract

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

Dates
Revised: 15 October 2011
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/ euclid.aspm/1540417812

Digital Object Identifier
doi:10.2969/aspm/06310001

Mathematical Reviews number (MathSciNet)
MR3051237

Zentralblatt MATH identifier
1321.19001

#### Citation

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. https://projecteuclid.org/euclid.aspm/1540417812