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Weierstrass preparation and algebraic invariants

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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.

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Acknowledgments

This work was done in part while the authors were in residence at the Banff International Research Station. We thank BIRS for its hospitality and resources, which helped advance the research in this paper. David Harbater was supported in part by NSF grant DMS-0901164. Julia Hartmann was supported by the German Excellence Initiative via RWTH Aachen University and by the German National Science Foundation (DFG). Daniel Krashen was supported in part by NSA grant H98130-08-0109 and NSF grant DMS-1007462.

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104-6395, USA

    David Harbater

  2. Lehrstuhl für Mathematik (Algebra), RWTH Aachen University, 52056 , Aachen, Germany

    Julia Hartmann

  3. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Daniel Krashen

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  1. David Harbater
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  2. Julia Hartmann
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  3. Daniel Krashen
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Correspondence to David Harbater.

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Harbater, D., Hartmann, J. & Krashen, D. Weierstrass preparation and algebraic invariants. Math. Ann. 356, 1405–1424 (2013). https://doi.org/10.1007/s00208-012-0888-8

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