Refinements to patching and applications to field invariants

Abstract

We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results.

44 pages. We renumbered results to be consistent with the numbering in the published version. We added a needed hypothesis to Corollary 4.13 that was missing from the published version

References in corpus (3)

• Applications of patching to quadratic forms and central simple algebras
• Period-index and u-invariant questions for function fields over complete discretely valued fields
• The Pythagoras number and the $u$-invariant of Laurent series fields in several variables

Cited by in corpus (6)

• Local-global principle for reduced norms over function fields of p-adic curves
• Local-global principles for tori over arithmetic curves
• Local-Global Principle for Unitary Groups Over Function Fields of p-adic Curves
• Generalized period-index problem with an application to quadratic forms
• Bounding the Pythagoras number of a field by $2^n+1$
• Reduced norms of division algebras over complete discrete valuation fields of local-global type