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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 . ==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 ''Â'' is an "infinitesimal neighborhood" ''D'' = Spec ''Â'' of (''f''). The intersection of ''D''''f'' and Spec ''Â'' is a "punctured infinitesimal neighborhood" ''D''0 about (''f''), equal to Spec ''Â'' ⊗''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, : (Despite the notational temptation to write , 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 ''D''0, and since both restrictions are ultimately derived from ''M'', they are isomorphic: we have an isomorphism : 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 ''D''0 = ''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 : consistent with the isomorphism ''φ'': ''φ'' is equal to the composition : 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 M''f''(''A'') be the full subcategory of ''f''-regular modules. In this notation, we obtain a commutative diagram of categories (note M''f''(''A''''f'') = M(''A''''f'')): : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Beauville–Laszlo theorem」の詳細全文を読む スポンサード リンク
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