Although it has implications in algebraic geometry, the theorem is a local result and is stated in its most primitive form for commutative rings.
, meaning the completion of the A-module M at the ideal Af, unless A is noetherian and M is finitely-generated, the two are not in fact equal.
In geometric language, the Beauville–Laszlo theorem allows one to glue sheaves on a one-dimensional affine scheme over an infinitesimal neighborhood of a point.
The version of this statement that the authors found noteworthy concerns vector bundles: Theorem: Let X be an algebraic curve over a field k, x a k-rational smooth point on X with infinitesimal neighborhood D = Spec k[[t]], R a k-algebra, and r a positive integer.
The corollary follows from the theorem in that the triple is associated with the unique matrix which, viewed as a "transition function" over D0R between the trivial bundles over (X \ x)R and over DR, allows gluing them to form E, with the natural trivializations of the glued bundle then being identified with σ and τ.