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

Beauville, Arnaud; Laszlo, Yves

C. R. Acad. Sci., Paris, Sér. I 320, No. 3, 335-340 (1995).

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