Applications of Coding Theory to the Construction of Modular Lattices

Introduction. Let K be a field. Springer has proved that an anisotropic quadratic form over K is also anisotropic over any odd degree extension of K (see [31], [14]). If the characteristic of K is not 2, this implies that two nonsingular quadratic forms that become isomorphic over an extension of odd degree of K are already isomorphic over K (see [31]). In [27], Serre reformulated the latter Statement äs follows: if O is an orthogonal group over K, then the canonical map of Galois cohomology sets

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Forms in odd degree extensions and self-dual normal bases

A spectral sequence is constructed whose non-zero E 1 -terms are the Witt groups of the residue fields of a regular scheme X , arranged in Gersten–Witt complexes, and whose limit is the four global Witt groups of  X . This has several immediate consequences concerning purity for Witt groups of low-dimensional schemes. We also obtain an easy proof of the Gersten Conjecture in dimension smaller than 5. The Witt groups of punctured spectra of regular local rings are also computed.

http://archive.numdam.org/article/ASENS_2002_4_35_1_127_0.pdf

A Gersten–Witt spectral sequence for regular schemes

We study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, devissage and localization theorems – preparing the ground for the theory of higher Grothendieck-Witt groups of schemes as developed in [Sch08a] and [Sch08b]. No assumption on the characteristic is being made.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1865243309999767

H.-G Quebbemann

Papers

Quadratic and Hermitian Forms in Additive and Abelian Categories

Lattices with theta functions for G(√2) and linear codes