Exalcomm Explained

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

and let all algebras be algebras over it. 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.

Definition

} 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

\begin{matrix} Spec\left(

	k[x,y]
	(y2-x3)

\right)&\to&Spec\left(

	k[x,y][\varepsilon]
	(y2-x3+\varepsilon)

\right)\\ \downarrow&&\downarrow\\ Spec(k)&\to&Spec(k[\varepsilon]) \end{matrix}

has the associated square-zero extension

0\to(\varepsilon)\to

	k[x,y][\varepsilon]
	(y2-x3+\varepsilon)

\to

	k[x,y]
	(y2-x3)

\to0

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^[1] follows from first defining a category of extensions

} over a topos (or just the category of commutative rings), then extracting a subcategory where a base ring }_A is fixed, and then using a functor }_A(B,-) \to \text 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 } \to \textsending a pair to a pair where is a -module.

Exal_A, Exal_A(B, –)

Then, there is an overcategory denoted

}_A (meaning there is a functor }_A \to ) where the objects are pairs , but the first ring is fixed, so morphisms are of the form There is a further reduction to another overcategory }_A(B,-) where morphisms are of the form

Exal_A(B,I)

Finally, the category

}_A(B,I) 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 }_A(B,I) under the functor lives in .

The isomorphism classes of objects has the structure of a

-module since }_A(B,I) 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

}_A(B,I) and which are useful.

Automorphisms

The group of automorphisms of an object

}_A(B,I)) can be identified with the automorphisms of the trivial extension (explicitly, we mean automorphisms compatible with both the inclusion and projection ). These are classified by the derivations module . Hence, the category }_A(B,I) 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

}_A(B,-) describing the extensions of , there is an isomorphism

\underline{Exal

}_A(B,I\oplus J) \cong \underline_A(B,I)\times \underline_A(B,J)

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

\underline{Exal

}_A(B,(\varepsilon_1) \oplus (\varepsilon_2)) \cong\underline_A(B,(\varepsilon_1))\times \underline_A(B,(\varepsilon_2))

where is the module of these two infinitesimals. 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

	1(X,T
H
	X) x

	1(X,T
H
	X)

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 (note taking as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphism

Exal_A(B,M)\xrightarrow{\simeq}

	1(L
Ext
	B/A

,M)

^{(theorem III.1.2.3)}

So, given a commutative square of ring morphisms

\begin{matrix} A'&\to&B'\\ \downarrow&&\downarrow\\ A&\to&B \end{matrix}

over there is a square

\begin{matrix} Exal_A(B,M)&\to&

	1
Ext
	B(L

_B/A,M)\\ \downarrow&&\downarrow\\ Exal_A'(B',M)&\to&

	1
Ext
	B'

(L_B'/A',M) \end{matrix}

whose horizontal arrows are isomorphisms and has the structure of a -module from the ring morphism.

See also

• Deformation theory
• Cotangent complex
• Picard stack

References

1. Book: Illusie, Luc. Complexe Cotangent et Deformations I. 151–168.

• Tangent Spaces and Obstruction Theories - Olsson