Exalcomm


In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M. 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". 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 DerA 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 undestand the construction of Exal, the notion of square-zero extensions must be defined. Fix a topos 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 we need to define what a square-zero extension actually is. Given a surjective morphism of -algebras it is called a square-zero extension if the kernel of has the property is the zero-ideal.

Remark

Note that the kernel can be equipped with a -module structure as follows: since is surjective, any has a lift to a , so for. Since any lift differs by an element in the kernel, and
because 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 numbers
has the associated square-zero extension
of -algebras.

From more general deformations

But, because the idea of square zero-extensions is more general, deformations over where will give examples of square-zero extensions.

Trivial square-zero extension

For a -module, there is a trivial square-zero extension given by where the product structure is given by
hence the associated square-zero extension is
where the surjection is the projection map forgetting.

Construction

The general abstract construction of Exal follows from first defining a category of extensions over a topos , then extracting a subcategory where a base ring is fixed, and then using a functor to get the module of commutative algebra extensions for a fixed.

General Exal

For this fixed topos, let be the category of pairs where is a surjective morphism of -algebras such that the kernel is square-zero, where morphisms are defined as commutative diagrams between. There is a functor
sending a pair to a pair where is a -module.

Exal''A,'' Exal''A''''(B, -)''

Then, there is an over category denoted where the objects are pairs, but the first ring is fixed, so morphisms are of the form
There is a further reduction to another over category where morphisms are of the form

Exal''A''(B,I)

Finally, the category has a fixed kernel of the square-zero extensions. Note that in, for a fixed, there is the subcategory where is a -module, so it is equivalent to. Hence, the image of under the functor lives in.
The isomorphism classes of objects has the structure of a -module since is a Picard stack, so the category can be turned into a module.

Structure of Exal''A''''(B, I)''

There are a few results on the structure of and which are useful.

Automorphisms

The group of automorphisms of an object can be identified with the automorphisms of the trivial extension. These are classified by the derivations module. Hence, the category 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 describing the extensions of, there is an isomorphism
It 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 where gives the isomorphism
where is the module of these two infinitesimals. In particular, when relating this to Kodaira-Spencer theory, and using the comparison with the contangent complex this means all such deformations are classified by
hence 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 over a topos , there is a functorial isomorphism
So, given a commutative square of ring morphisms
over there is a square
whose horizontal arrows are isomorphisms and has the structure of a -module from the ring morphism.