  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.
