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→Performance: Replaced average with worst. Average case can be less than n log n because average depends on input distribution and for certain pathological distributions (for example if at most 1 element is misplaced), there are sorts that do better.
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In [[commutative algebra]], '''André–Quillen cohomology''' is a theory of [[cohomology]] for [[commutative ring]]s which is closely related to the [[cotangent complex]]. The first three cohomology groups were introduced by {{harvtxt|Lichtenbaum|Schlessinger|1967}} and are sometimes called '''Lichtenbaum–Schlessinger functors''' ''T''<sup>0</sup>, ''T''<sup>1</sup>, ''T''<sup>2</sup>, and the higher groups were defined independently by M. André and by [[Daniel Quillen]] using methods of [[homotopy theory]]. It comes with a parallel homology theory called '''André–Quillen homology'''.


==Motivation==
Let ''A'' be a commutative ring, ''B'' be an ''A''-algebra, and ''M'' be a ''B''-module. André–Quillen cohomology is the derived functors of the [[derivation (abstract algebra)|derivation]] functor Der<sub>''A''</sub>(''B'', ''M''). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings {{nowrap|''A'' &rarr; ''B'' &rarr; ''C''}} and a ''C''-module ''M'', there is a three-term [[exact sequence]] of derivation modules:
:<math>0 \to \operatorname{Der}_B(C, M) \to \operatorname{Der}_A(C, M) \to \operatorname{Der}_A(B, M).</math>
This term can be extended to a six-term exact sequence using the functor [[Exalcomm]] of extensions of commutative algebras and a nine-term exact sequence using the [[Lichtenbaum–Schlessinger functor]]s. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.


==Definition==
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Let ''B'' be an ''A''-algebra, and let ''M'' be a ''B''-module. Let ''P'' be a simplicial cofibrant ''A''-algebra resolution of ''B''. André notates the ''q''th cohomology group of ''B'' over ''A'' with coefficients in ''M'' by {{nowrap|H<sup>''q''</sup>(''A'', ''B'', ''M'')}}, while Quillen notates the same group as {{nowrap|D<sup>''q''</sup>(''B''/''A'', ''M'')}}. The ''q''th '''André–Quillen cohomology group''' is:
:<math>D^q(B/A, M) = H^q(A, B, M) \stackrel{\text{def}}{=} H^q(\operatorname{Der}_A(P, M)).</math>
The ''q''th '''André–Quillen homology group''' is:
:<math>D_q(B/A, M) = H_q(A, B, M) \stackrel{\text{def}}{=} H_q(\Omega_{P/A} \otimes_B M).</math>
 
Let {{nowrap|L<sub>''B''/''A''</sub>}} denote the relative [[cotangent complex]] of ''B'' over ''A''. Then we have the formulas:
:<math>D^q(B/A, M) = H^q(\operatorname{Hom}_B(L_{B/A}, M)),</math>
:<math>D_q(B/A, M) = H_q(L_{B/A} \otimes_B M).</math>
 
==References==
*{{Citation | last1=André | first1=M. | title=Homologie des Algèbres Commutatives | series=Grundlehren der mathematischen Wissenschaften | volume=206 | publisher=[[Springer-Verlag]] | year=1974}}
*{{Citation | last1=Lichtenbaum | first1=Stephen |author1-link=Stephen Lichtenbaum | last2=Schlessinger | first2=M. | title=The cotangent complex of a morphism | url=http://www.jstor.org/stable/1994516 | id={{MR|0209339}} | year=1967 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=128 | pages=41–70}}
* {{citation| last1=Quillen | first1=Daniel G. | title=Homology of commutative rings|url=http://chromotopy.org/paste/quillen.djvu|series=unpublished notes}}
*{{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}}
*{{Citation | last1=Weibel | first1=Charles A. | title=An introduction to homological algebra | url=http://books.google.com/books?id=flm-dBXfZ_gC | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-43500-0 | id={{MR|1269324}} | year=1994 | volume=38}}
 
{{DEFAULTSORT:Andre-Quillen cohomology}}
[[Category:Commutative algebra]]
[[Category:Homotopy theory]]

Revision as of 10:14, 13 February 2014


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