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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.
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| ==Motivation==
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| Suppose that ''X'' and ''Y'' are [[algebraic variety|algebraic varieties]] and that {{nowrap|''f'' : ''X'' → ''Y''}} is a morphism between them. The cotangent complex of ''f'' is a more universal version of the relative [[Kähler differentials]] Ω<sub>''X''/''Y''</sub>. 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 {{nowrap|''g'' : ''Y'' → ''Z''}} is another morphism, then there is an exact sequence
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| :<math>f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0.</math> | |
| 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 functor]]s ''T''<sup>''i''</sup> and [[imperfection module]]s. Most of these were motivated by [[deformation theory]].
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| 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.
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| 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
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| :<math>I/I^2 \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to 0.</math>
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| 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<sub>''X''/''Y''</sub> 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.<ref>{{Harvard citations|last = Grothendieck|year = 1967|loc = Proposition 17.2.5|nb = yes}}</ref> This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term.
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| ==Early work on cotangent complexes==
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| The cotangent complex dates back at least to SGA 6 VIII 2, where [[Pierre Berthelot]] gave a definition when ''f'' is a ''smoothable'' morphism, meaning there is a scheme ''V'' and morphisms {{nowrap|''i'' : ''X'' → ''V''}} and {{nowrap|''h'' : ''V'' → ''Y''}} such that {{nowrap|''f'' {{=}} ''hi''}}, ''i'' is a closed immersion, and ''h'' is a smooth morphism. (For example, all projective morphisms are smoothable, since ''V'' can be taken to be a projective bundle over ''Y''.) In this case, he defines the cotangent complex of ''f'' as an object in the [[derived category]] of [[coherent sheaf|coherent sheaves]] ''X'' as follows:
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| *<math>L^{X/Y}_0 = i^*\Omega_{V/Y},</math>
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| *If ''J'' is the ideal of ''X'' in ''V'', then <math>L^{X/Y}_1 = J/J^2 = i^*J</math>,
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| *<math>L^{X/Y}_i = 0</math> for all other ''i'',
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| *The differential <math>L^{X/Y}_1 \to L^{X/Y}_0</math> is the pullback along ''i'' of the inclusion of ''J'' in the structure sheaf <math>\mathcal{O}_V</math> of ''V'' followed by the universal derivation <math>d : \mathcal{O}_V \to \Omega_{V/Y}</math>.
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| *All other differentials are zero.
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| Berthelot proves that this definition is independent of the choice of ''V''<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.2|nb = yes}}</ref> and that for a smoothable complete intersection morphism, this complex is perfect.<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.4|nb = yes}}</ref> Furthermore, he proves that if {{nowrap|''g'' : ''Y'' → ''Z''}} is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an [[exact triangle]]
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| :<math>\mathbf{L}f^*L^{Y/Z}_\bullet \to L^{X/Z}_\bullet \to L^{X/Y}_\bullet \to \mathbf{L}f^*L^{Y/Z}_\bullet[1].</math>
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| ==The definition of the cotangent complex==
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| The correct definition of the cotangent complex begins in the [[homotopic algebra|homotopical setting]]. Quillen and André worked with the [[simplicial set#Simplicial objects|simplicial]] commutative rings, while Illusie worked with simplicial ringed [[topos|topoi]]. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that ''A'' and ''B'' are [[simplicial ring]]s and that ''B'' is an ''A''-algebra. Choose a resolution {{nowrap|''r'' : ''P''<sup>•</sup> → ''B''}} of ''B'' by simplicial free ''A''-algebras. Applying the Kähler differential functor to ''P''<sup>•</sup> produces a simplicial ''B''-module. The total complex of this simplicial object is the '''cotangent complex''' ''L''<sup>''B''/''A''</sup>. The morphism ''r'' induces a morphism from the cotangent complex to Ω<sub>''B''/''A''</sub> called the '''augmentation map'''. In the homotopy category of simplicial ''A''-algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler differential functor.
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| Given a commutative square as follows:
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| :[[File:Commutative square.svg]]
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| there is a morphism of cotangent complexes {{nowrap|''L''<sup>''B''/''A''</sup> ⊗<sub>''B''</sub> ''D'' → ''L''<sup>''D''/''C''</sup>}} which respects the augmentation maps. This map is constructed by choosing a free simplicial ''C''-algebra resolution of ''D'', say {{nowrap|''s'' : ''Q''<sup>•</sup> → ''D''}}. Because ''P''<sup>•</sup> is a free object, the composite ''hr'' can be lifted to a morphism {{nowrap|''P''<sup>•</sup> → ''Q''<sup>•</sup>}}. Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms {{nowrap|''A'' → ''B'' → ''C''}}, this produces the sequence
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| :<math>L^{B/A} \otimes_B C \to L^{C/A} \to L^{C/B}.</math>
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| There is a connecting homomorphism <math>L^{C/B} \to (L^{B/A} \otimes_B C)[1]</math> which turns this sequence into an exact triangle.
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| The cotangent complex can also be defined in any combinatorial [[model category]] ''M''. Suppose that <math>f\colon A\rightarrow B</math> is a morphism in ''M''. The cotangent complex <math>L^f</math> (or <math>L^{B/A}</math>) is an object in the category of spectra in <math>M_{B//B}</math>. A pair of composable morphisms <math>A\xrightarrow{f} B\xrightarrow{g} C</math> induces an exact triangle in the homotopy category, <math>L^{B/A}\otimes_BC\rightarrow L^{C/A}\rightarrow L^{C/B}\rightarrow (L^{B/A}\otimes_BC)[1]</math>.
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| ==Properties of the cotangent complex==
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| ===Flat base change===
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| Suppose that ''B'' and ''C'' are ''A''-algebras such that {{nowrap|Tor<sup>''A''</sup><sub>''q''</sub>(''B'', ''C'') {{=}} 0}} for all {{nowrap|''q'' > 0}}. Then there are quasi-isomorphisms<ref>{{Harvard citations|last = Quillen|year = 1970|loc = Theorem 5.3|nb = yes}}</ref>
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| :<math>L^{B \otimes_A C/C} \cong B \otimes_A L^{C/A},</math>
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| :<math>L^{B \otimes_A C/A} \cong (L^{B/A} \otimes_A C) \oplus (B \otimes_A L^{C/A}).</math>
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| If ''C'' is a flat ''A''-algebra, then the condition that {{nowrap|Tor<sup>''A''</sup><sub>''q''</sub>(''B'', ''C'')}} vanishes for {{nowrap|''q'' > 0}} is automatic. The first formula then proves that the construction of the cotangent complex is local on the base in the [[flat topology]].
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| ===Vanishing properties===
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| Let {{nowrap|''f'' : ''A'' → ''B''}}. Then:<ref>{{Harvard citations|last = Quillen|year = 1970|loc = Theorem 5.4|nb = yes}}</ref><ref>{{Harvard citations|last = Quillen|year = 1970|loc = Corollary 6.14|nb = yes}}</ref>
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| *If ''B'' is a [[localization of a ring|localization]] of ''A'', then {{nowrap|''L''<sup>''B''/''A''</sup> {{=}} 0}}.
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| *If ''f'' is an [[étale morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup> {{=}} 0}}.
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| *If ''f'' is a [[smooth morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup>}} is quasi-isomorphic to Ω<sub>''B''/''A''</sub>. In particular, it has [[projective dimension]] zero.
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| *If ''f'' is a [[local complete intersection morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup>}} has projective dimension at most one.
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| *If ''A'' is Noetherian, {{nowrap|''B'' {{=}} ''A''/''I''}}, and ''I'' is generated by a regular sequence, then <math>I/I^2</math> is a [[projective module]] and ''L''<sup>''B''/''A''</sup> is quasi-isomorphic to <math>I/I^2[1]</math>.
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| ==Examples==
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| *Let ''X'' be smooth over ''S''. Then the cotangent complex is Ω<sub>''X''/''S''</sub>. In Berthelot's framework, this is clear by taking {{nowrap|''V'' {{=}} ''X''}}. In general, étale locally on ''S'', ''X'' is a finite dimensional affine space and the morphism from ''X'' to ''S'' is projection, so we may reduce to the situation where {{nowrap|''S'' {{=}} Spec ''A''}} and {{nowrap|''X'' {{=}} Spec ''A''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}}. We can take the resolution of {{nowrap|''A''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.
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| *Let ''X'' and ''Y'' be smooth over ''S'', and assume that {{nowrap|''i'' : ''X'' → ''Y''}} is a closed embedding. Using the exact triangle corresponding to the morphisms {{nowrap|''X'' → ''Y'' → ''S''}}, we may determine the cotangent complex ''L''<sub>''X''/''Y''</sub>. To do this, note that by the previous example, the cotangent complexes ''L''<sub>''X''/''S''</sub> and ''L''<sub>''Y''/''S''</sub> consist of the Kähler differentials Ω<sub>''X''/''S''</sub> and Ω<sub>''Y''/''S''</sub> in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that ''L''<sub>''X''/''Y''</sub> is nonzero only in the first degree, and in that degree, it is the kernel of the map {{nowrap|''i''<sup>*</sup>Ω<sub>''Y''/''S''</sub> → Ω<sub>''X''/''S''</sub>}}. This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, ''L''<sub>''X''/''Y''</sub> is the conormal bundle of ''X'' in ''Y''.
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| ==See also==
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| *[[André–Quillen cohomology]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=André | first1=M. | title=Homologie des Algèbres Commutatives | series=Grundlehren der mathematischen Wissenschaften | volume=206 | publisher=[[Springer-Verlag]] | year=1974}}
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| *{{Citation
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| | last = Berthelot
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| | first = Pierre
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| | authorlink = Pierre Berthelot (mathematician)
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| | coauthors = [[Alexandre Grothendieck]], [[Luc Illusie]], eds.
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| | title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
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| | year = 1971
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | location = Berlin; New York
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| | language = French
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| | pages = xii+700
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| | nopp = true
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| }}
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| *{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1967__32_ | year=1967 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=32 | pages=5–361 | doi=10.1007/BF02732123}}
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| *{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | title=Catégories cofibrées additives et complexe cotangent relatif | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=French | series=Lecture Notes in Mathematics '''79''' | isbn=978-3-540-04248-8 | date=01/07/1969 }}
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| *{{Citation | last1=Harrison | first1=D. K. | title=Commutative algebras and cohomology | journal=Transactions of the American Mathematical Society | volume=104 | year=1962 | pages=191–204 | doi=10.2307/1993575 | jstor=1993575 | issue=2 | publisher=American Mathematical Society}}
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| *{{Citation | last1=Illusie | first1=Luc | author1-link=Luc Illusie | title=Complexe Cotangent et Déformations I | origyear=1971 | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=French | series=Lecture Notes in Mathematics '''239''' | isbn=978-3-540-05686-7 | year=2009}}
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| *{{Citation | last1=Lichtenbaum | last2=Schlessinger | title=The cotangent complex of a morphism | journal=Transactions of the American Mathematical society | issue=128 | year=1967 | pages=41–70}}
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| *{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=On the (co-)homology of commutative rings | series=Proc. Symp. Pure Mat. | volume=XVII | publisher=[[American Mathematical Society]] | year=1970}}
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| {{DEFAULTSORT:Cotangent Complex}}
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| [[Category:Algebraic geometry]]
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| [[Category:Category theory]]
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| [[Category:Homotopy theory]]
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