André–Quillen Cohomology
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In
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, André–Quillen cohomology is a theory of
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
for commutative rings which is closely related to the
cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic o ...
. The first three cohomology groups were introduced by and are sometimes called Lichtenbaum–Schlessinger functors ''T''0, ''T''1, ''T''2, and the higher groups were defined independently by and 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. The André–Quillen cohomology groups are the derived functors of the
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
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 rings and a ''C''-module ''M'', there is a three-term
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of derivation modules: :0 \to \operatorname_B(C, M) \to \operatorname_A(C, M) \to \operatorname_A(B, M). This term can be extended to a six-term exact sequence using the functor
Exalcomm In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcomm''k''(''R'',''M'') are isomorphism classes of commutative ''k''-algebras ''E'' with a homomorphism onto the '' ...
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.


Definition

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 , while Quillen notates the same group as . The ''q''th André–Quillen cohomology group is: :D^q(B/A, M) = H^q(A, B, M) \stackrel H^q(\operatorname_A(P, M)). Let denote the relative
cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic o ...
of ''B'' over ''A''. Then we have the formulas: :D^q(B/A, M) = H^q(\operatorname_B(L_, M)), :D_q(B/A, M) = H_q(L_ \otimes_B M).


See also

*
Cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic o ...
* Deformation Theory *
Exalcomm In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcomm''k''(''R'',''M'') are isomorphism classes of commutative ''k''-algebras ''E'' with a homomorphism onto the '' ...


References

* * * * *


Generalizations


André–Quillen cohomology of commutative S-algebras
* Homology and Cohomology of E-infinity ring spectra {{DEFAULTSORT:Andre-Quillen cohomology Commutative algebra Homotopy theory Cohomology theories