Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1995).
Although it has implications in algebraic geometry, the theorem is a local result and is stated in its most primitive form for commutative rings. If A is a ring and f is a nonzero element of A, then we can form two derived rings: the localization at f, Af, and the completion at Af, Â; both are A-algebras. In the following we assume that f is a non-zero divisor. Geometrically, A is viewed as a scheme X = Spec A and f as a divisor (f) on Spec A; then Af is its complement Df = Spec Af, the principal open set determined by f, while  is an "infinitesimal neighborhood" D = Spec  of (f). The intersection of Df and Spec  is a "punctured infinitesimal neighborhood" D0 about (f), equal to Spec  ⊗AAf = Spec Âf.
Suppose now that we have an A-module M; geometrically, M is a sheaf on Spec A, and we can restrict it to both the principal open set Df and the infinitesimal neighborhood Spec Â, yielding an Af-module F and an Â-module G. Algebraically,
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