In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
. 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
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
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
. 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