Weierstrass preparation and algebraic invariants

TL;DR

This paper extends the Weierstrass Preparation Theorem to normal algebraic curves over complete discrete valuation rings, enabling new applications in algebraic invariants and division algebra problems.

Contribution

It introduces a generalized Weierstrass Preparation Theorem for curves beyond the projective line, utilizing patching and matrix factorization techniques.

Findings

Generalized Weierstrass Preparation for algebraic curves

Applications to u-invariants and period-index problems

Use of patching methods and matrix factorizations

Abstract

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan's Lemma.