Incommutative algebra,André–Quillen cohomology is a theory ofcohomology forcommutative rings which is closely related to thecotangent complex. The first three cohomology groups were introduced byStephen Lichtenbaum andMichael Schlessinger (1967) and are sometimes calledLichtenbaum–Schlessinger functorsT⁰,T¹,T², and the higher groups were defined independently byMichel André (1974) andDaniel Quillen (1970) using methods ofhomotopy theory. It comes with a parallel homology theory calledAndré–Quillen homology.

LetA be a commutative ring,B be anA-algebra, andM be aB-module. The André–Quillen cohomology groups are the derived functors of thederivation functor Der_A(B,M). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative ringsA →B →C and aC-moduleM, there is a three-termexact sequence of derivation modules:

${\displaystyle 0\to {\mathrm{Der}}_B(C,M)\to {\mathrm{Der}}_A(C,M)\to {\mathrm{Der}}_A(B,M).}$

This term can be extended to a six-term exact sequence using the functorExalcomm of extensions of commutative algebras and a nine-term exact sequence using the Lichtenbaum–Schlessinger functors. 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.

LetB be anA-algebra, and letM be aB-module. LetP be a simplicial cofibrantA-algebra resolution ofB. André notates theqth cohomology group ofB overA with coefficients inM byH^q(A,B,M), while Quillen notates the same group asD^q(B/A,M). TheqthAndré–Quillen cohomology group is:

${\displaystyle D^q(B/A,M)=H^q(A,B,M)\stackrel{\text{def}}{=}{H}^q({\mathrm{Der}}_A(P,M)).}$

LetL_B/A denote the relativecotangent complex ofB overA. Then we have the formulas:

${\displaystyle D^q(B/A,M)=H^q({\mathrm{Hom}}_B(L_{B/A},M)),}$

${\displaystyle D_q(B/A,M)=H_q(L_{B/A}{\otimes }_{B}M).}$

• Cotangent complex
• Deformation Theory
• Exalcomm

• André, Michel (1974),Homologie des Algèbres Commutatives, Grundlehren der mathematischen Wissenschaften, vol. 206,Springer-Verlag
• Lichtenbaum, Stephen;Schlessinger, Michael (1967), "The cotangent complex of a morphism",Transactions of the American Mathematical Society,128 (1):41–70,doi:10.2307/1994516,ISSN 0002-9947,JSTOR 1994516,MR 0209339
• Quillen, Daniel G.,Homology of commutative rings, unpublished notes, archived fromthe original on April 20, 2015
• Quillen, Daniel (1970),On the (co-)homology of commutative rings, Proc. Symp. Pure Mat., vol. XVII,American Mathematical Society
• Weibel, Charles A. (1994),An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38,Cambridge University Press,doi:10.1017/CBO9781139644136,ISBN 978-0-521-43500-0,MR 1269324

• André–Quillen cohomology of commutative S-algebras
• Homology and Cohomology of E-infinity ring spectra

• Commutative algebra
• Homotopy theory
• Cohomology theories

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