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).

The theorem

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, A_f, 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 A_f is its complement D_f = Spec A_f, the principal open set determined by f, while  is an "infinitesimal neighborhood" D = Spec  of (f). The intersection of D_f and Spec  is a "punctured infinitesimal neighborhood" D⁰ about (f), equal to Spec  ⊗_A A_f = Spec Â_f.

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 D_f and the infinitesimal neighborhood Spec Â, yielding an A_f-module F and an Â-module G. Algebraically,

\( F = M \otimes_A A_f = M_f \qquad G = M \otimes_A \hat{A}. \)

(Despite the notational temptation to write \( G = \widehat{M} \), 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. This phenomenon is the main reason that the theorem bears the names of Beauville and Laszlo; in the noetherian, finitely-generated case, it is, as noted by the authors, a special case of Grothendieck's faithfully flat descent.) F and G can both be further restricted to the punctured neighborhood D0, and since both restrictions are ultimately derived from M, they are isomorphic: we have an isomorphism

\( \phi \colon G_f \xrightarrow{\sim} F \otimes_{A_f} \hat{A}_f = F \otimes_A \hat{A}. \)

Now consider the converse situation: we have a ring A and an element f, and two modules: an A_f-module F and an Â-module G, together with an isomorphism φ as above. Geometrically, we are given a scheme X and both an open set D_f and a "small" neighborhood D of its closed complement (f); on D_f and D we are given two sheaves which agree on the intersection D⁰ = D_f ∩ D. If D were an open set in the Zariski topology we could glue the sheaves; the content of the Beauville–Laszlo theorem is that, under one technical assumption on f, the same is true for the infinitesimal neighborhood D as well.

Theorem: Given A, f, F, G, and φ as above, if G has no f-torsion, then there exist an A-module M and isomorphisms

\( \alpha \colon M_f \xrightarrow{\sim} F \qquad \beta \colon M \otimes_A \hat{A} \xrightarrow{\sim} G \)

consistent with the isomorphism φ: φ is equal to the composition

\( G_f = G \otimes_A A_f \xrightarrow{\beta^{-1} \otimes 1} M \otimes_A \hat{A} \otimes_A A_f = M_f \otimes_A \hat{A} \xrightarrow{\alpha \otimes 1} F \otimes_A \hat{A}. \)

The technical condition that G has no f-torsion is referred to by the authors as "f-regularity". In fact, one can state a stronger version of this theorem. Let M(A) be the category of A-modules (whose morphisms are A-module homomorphisms) and let Mf(A) be the full subcategory of f-regular modules. In this notation, we obtain a commutative diagram of categories (note Mf(Af) = M(Af)):

\( \begin{array}{ccc} \mathbf{M}_f(A) & \longrightarrow & \mathbf{M}_f(\hat{A}) \\ \downarrow & & \downarrow \\ \mathbf{M}(A_f) & \longrightarrow & \mathbf{M}(\hat{A}_f) \end{array} \)

in which the arrows are the base-change maps; for example, the top horizontal arrow acts on objects by M → M ⊗A Â.

Theorem: The above diagram is a cartesian diagram of categories.

Global version

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. Since sheaves have a "local character" and since any scheme is locally affine, the theorem admits a global statement of the same nature. 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. Then the category Vect_r(X_R) of rank-r vector bundles on the curve X_R = X ×_{Spec k} Spec R fits into a cartesian diagram:

\( \begin{array}{ccc} \mathbf{Vect}_r(X_R) & \longrightarrow & \mathbf{Vect}_r(D_R) \\ \downarrow & & \downarrow \\ \mathbf{Vect}_r((X \setminus x)_R) & \longrightarrow & \mathbf{Vect}_r(D_R^0) \end{array} \)

This entails a corollary stated in the paper:

Corollary:

With the same setup, denote by Triv(X_R) the set of triples (E, τ, σ), where E is a vector bundle on X_R, τ is a trivialization of E over (X \ x)_R (i.e., an isomorphism with the trivial bundle O_{(X - x)R}), and σ a trivialization over D_R. Then the maps in the above diagram furnish a bijection between Triv(X_R) and GL_r(R((t))) (where R((t)) is the formal Laurent series ring).

The corollary follows from the theorem in that the triple is associated with the unique matrix which, viewed as a "transition function" over D⁰_R between the trivial bundles over (X \ x)_R and over D_R, allows gluing them to form E, with the natural trivializations of the glued bundle then being identified with σ and τ. The importance of this corollary is that it shows that the affine Grassmannian may be formed either from the data of bundles over an infinitesimal disk, or bundles on an entire algebraic curve.

References

Beauville, Arnaud; Laszlo, Yves (1995), "Un lemme de descente" (PDF), Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 320 (3): 335–340, ISSN 0764-4442, retrieved 2008-04-08.

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