More precisely, the elements of Exalcommk(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).
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 (Grothendieck & Dieudonné 1964, 20.2.3.1) where DerA(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.
In order to understand the construction of Exal, the notion of square-zero extensions must be defined.
Note that the topos of a point gives the special case of commutative rings, so the topos hypothesis can be ignored on a first reading.
it is called a square-zero extension if the kernel
in the kernel, and because the ideal is square-zero, this module structure is well-defined.
Square-zero extensions are a generalization of deformations over the dual numbers.
For example, a deformation over the dual numbers
But, because the idea of square zero-extensions is more general, deformations over
will give examples of square-zero extensions.
where the product structure is given by hence the associated square-zero extension is where the surjection is the projection map forgetting
The general abstract construction of Exal[1] follows from first defining a category of extensions
to get the module of commutative algebra extensions
is square-zero, where morphisms are defined as commutative diagrams between
is fixed, so morphisms are of the form There is a further reduction to another overcategory
where morphisms are of the form Finally, the category
has a fixed kernel of the square-zero extensions.
The isomorphism classes of objects has the structure of a
is a Picard stack, so the category can be turned into a module
can be identified with the automorphisms of the trivial extension
In fact, this could also be interpreted as a Gerbe since this is a group acting on a stack.
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.
In particular, when relating this to Kodaira-Spencer theory, and using the comparison with the cotangent complex (given below) this means all such deformations are classified by
The cotangent complex contains all of the information about a deformation problem, and it is a fundamental theorem that given a morphism of rings
as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphism
[1](theorem III.1.2.3)So, given a commutative square of ring morphisms