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In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. ==Definitions== Λ is a complete Noetherian local ring with residue field ''k'', and ''C'' is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field ''k''. A small extension in ''C'' is a morphism ''Y''→''Z'' in ''C'' that is surjective with kernel a 1-dimensional vector space over ''k''. A functor is called representable if it is of the form ''h''''X'' where ''h''''X''(''Y'')=hom(''X'',''Y'') for some ''X'', and is called pro-representable if it is of the form ''Y''→lim hom(''X''''i'',''Y'') for a filtered direct limit over ''i'' in some filtered ordered set. A morphism of functors ''F''→''G'' from ''C'' to sets is called smooth if whenever ''Y''→''Z'' is an epimorphism of ''C'', the map from ''F''(''Y'') to ''F''(''Z'')×''G''(''Z'')''G''(''Y'') is surjective. This definition is closely related to the notion of a formally smooth morphism of schemes. If in addition the map between the tangent spaces of ''F'' and ''G'' is an isomorphism, then ''F'' is called a hull of ''G''. 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「Schlessinger's theorem」の詳細全文を読む スポンサード リンク
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