A descent lemma. (Un lemme de descente.)(French. Abridged English version)Zbl 0852.13005

Let $$A$$ be a commutative ring, $$f$$ a non zero-divisor in $$A$$, $$\widehat A$$ the $$(f)$$-adic completion of $$A$$ and $$A_f$$ and $$\widehat A_f$$ the localizations at $$f$$. The authors’ main result asserts that given an $$A_f$$-module $$F$$, an $$\widehat A$$-module $$G$$ on which multiplication by $$f$$ is injective, and an $$\widehat A_f$$-isomorphism $$\varphi : \widehat A \otimes_AF \to G$$ the triple $$(F,G,\varphi)$$ comes from an $$A$$-module $$M$$, so that $$M_f = F$$, $$\widehat A \otimes_AM = G$$ and $$\varphi$$ is the obvious isomorphism. – The authors further note that the theorem easily generalizes to the global situation, and deduce the following corollary:
Let $$X$$ be an algebraic curve over the field $$k$$, let $$p$$ be a smooth rational point on $$X$$, let $$z$$ be a local coordinate at $$p$$ and let $$R$$ be a $$k$$-algebra. Then there is a functional bijection between the set of isomorphism classes of triples $$(E, \tau, \sigma)$$ of rank $$r$$ vector bundles on $$X_R = X \times_k \text{Spec} (R)$$ with trivializations $$\tau$$ over $$(X - p)_R$$ and $$\sigma$$ over $$\text{Spec} (R[[z]])$$; and the group $$GL_r (R((z)))$$.
Reviewer: A.R.Magid (Norman)

MSC:

 13B30 Rings of fractions and localization for commutative rings 13J10 Complete rings, completion 14H60 Vector bundles on curves and their moduli

Keywords:

completion; localizations; vector bundles