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 "k"-algebra "R" whose kernel is the "R"-module "M" (with all pairs of elements in "M" having product 0). Note that some authors use Exal as the same functor. There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop. and Exalcotop that take a topology into account.
"Exalcomm" is an abbreviation for "COMMutative ALgebra EXtension" (or rather for the corresponding French phrase). It was introduced by .
Exalcomm is one of the André–Quillen cohomology groups and one of the Lichtenbaum–Schlessinger functors.
Given homomorphisms of commutative rings "A" → "B" → "C" and a "C"-module "L" there is an exact sequence of "A"-modules 
where Der"A"("B","L") is the module of derivations of the "A"-algebra "B" with values in "L". 
This sequence can be extended further to the right using André–Quillen cohomology.
Square-zero extensions.
In order to understand the construction of Exal, the notion of square-zero extensions must be defined. Fix a topos formula_2 and let all algebras be algebras over it. Note that the topos of a point gives the special case of commutative rings, so ignoring the topos hypothesis can be ignored on a first reading.
Definition.
In order to define the category formula_3 we need to define what a square-zero extension actually is. Given a surjective morphism of formula_4-algebras formula_5 it is called a square-zero extension if the kernel formula_6 of formula_7 has the property formula_8 is the zero-ideal.
Remark.
Note that the kernel can be equipped with a formula_9-module structure as follows: since formula_7 is surjective, any formula_11 has a lift to a formula_12 , so formula_13 for formula_14. Since any lift differs by an element formula_15 in the kernel, andformula_16because the ideal is square-zero, this module structure is well-defined.
Examples.
From deformations over the dual numbers.
Square-zero extensions are a generalization of deformations over the dual numbers. For example, a deformation over the dual numbersformula_17has the associated square-zero extensionformula_18of formula_19-algebras.
From more general deformations.
But, because the idea of square zero-extensions is more general, deformations over formula_20 where formula_21 will give examples of square-zero extensions.
Trivial square-zero extension.
For a formula_9-module formula_23, there is a trivial square-zero extension given by formula_24 where the product structure is given byformula_25hence the associated square-zero extension isformula_26where the surjection is the projection map forgetting formula_23.
Construction.
The general abstract construction of Exal follows from first defining a category of extensions formula_3 over a topos formula_2 (or just the category of commutative rings), then extracting a subcategory where a base ring formula_4 formula_31 is fixed, and then using a functor formula_32 to get the module of commutative algebra extensions formula_33 for a fixed formula_34.
General Exal.
For this fixed topos, let formula_3 be the category of pairs formula_36 where formula_37 is a surjective morphism of formula_4-algebras such that the kernel formula_6 is square-zero, where morphisms are defined as commutative diagrams between formula_40. There is a functorformula_41sending a pair formula_36 to a pair formula_43 where formula_6 is a formula_9-module. 
Exal"A," Exal"A""(B, -)".
Then, there is an over category denoted formula_31 (meaning there is a functor formula_47) where the objects are pairs formula_36, but the first ring formula_4 is fixed, so morphisms are of the form formula_50There is a further reduction to another over category formula_51 where morphisms are of the formformula_52
Exal"A"(B,I).
Finally, the category formula_53 has a fixed kernel of the square-zero extensions. Note that in formula_54, for a fixed formula_55, there is the subcategory formula_43 where formula_6 is a formula_9-module, so it is equivalent to formula_59. Hence, the image of formula_53 under the functor formula_61 lives in formula_59.
The isomorphism classes of objects has the structure of a formula_9-module since formula_53 is a Picard stack, so the category can be turned into a module formula_65.
Structure of Exal"A""(B, I)".
There are a few results on the structure of formula_53 and formula_65 which are useful.
Automorphisms.
The group of automorphisms of an object formula_68 can be identified with the automorphisms of the trivial extension formula_69. These are classified by the derivations module formula_70. Hence, the category formula_53 is a torsor. In fact, this could also be interpreted as a Gerbe since this is a group acting on a stack.
Composition of extensions.
There is another useful result about the categories formula_51 describing the extensions of formula_73, there is an isomorphismformula_74It can be interpreted as saying the square-zero extension from a deformation in two directions can be decomposed into a pair of square-zero extensions, each in the direction of one of the deformations.
Application.
For example, the deformations given by infinitesimals formula_75 where formula_76 gives the isomorphismformula_77where formula_6 is the module of these two infinitesimals. In particular, when relating this to Kodaira-Spencer theory, and using the comparison with the contangent complex (given below) this means all such deformations are classified byformula_79hence they are just a pair of first order deformations paired together.
Relation with the cotangent complex.
The cotangent complex contains all of the information about a deformation problem, and it is a fundamental theorem that given a morphism of rings formula_80 over a topos formula_2 (note taking formula_2 as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphismformula_83(theorem III.1.2.3)So, given a commutative square of ring morphismsformula_84over formula_2 there is a squareformula_86whose horizontal arrows are isomorphisms and formula_23 has the structure of a formula_88-module from the ring morphism.