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In mathematics the cotangent complex is roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases. ==Motivation== Suppose that ''X'' and ''Y'' are algebraic varieties and that is a morphism between them. The cotangent complex of ''f'' is a more universal version of the relative Kähler differentials Ω''X''/''Y''. The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If ''Z'' is another variety, and if is another morphism, then there is an exact sequence : In some sense, therefore, relative Kähler differentials are a right exact functor. (Literally this is not true, however, because the category of algebraic varieties is not an abelian category, and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the Lichtenbaum–Schlessinger functors ''T''''i'' and imperfection modules. Most of these were motivated by deformation theory. This sequence is exact on the left if the morphism ''f'' is smooth. If Ω admitted a first derived functor, then exactness on the left would imply that the connecting homomorphism vanished, and this would certainly be true if the first derived functor of ''f'', whatever it was, vanished. Therefore a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials. Another natural exact sequence related to Kähler differentials is the conormal exact sequence. If ''f'' is a closed immersion with ideal sheaf ''I'', then there is an exact sequence : This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of ''f'', and the relative differentials Ω''X''/''Y'' have vanished because a closed immersion is formally unramified. If ''f'' is the inclusion of a smooth subvariety, then this sequence is a short exact sequence. This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term. 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「cotangent complex」の詳細全文を読む スポンサード リンク
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