# 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 Grothendieck & Dieudonné (1964, 18.4.2).
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)
    ${\begin{aligned}0\rightarrow \;&\operatorname {Der} _{B}(C,L)\rightarrow \operatorname {Der} _{A}(C,L)\rightarrow \operatorname {Der} _{A}(B,L)\rightarrow \\&\operatorname {Exalcomm} _{B}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(B,L)\end{aligned}}$
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 $T$ 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

In order to define the category ${\underline {\text{Exal}}}$ we need to define what a square-zero extension actually is. Given a surjective morphism of $A$-algebras $p:E\to B$ it is called a square-zero extension if the kernel $I$ of $p$ has the property $I^{2}=(0)$ is the zero ideal.

#### Remark

Note that the kernel can be equipped with a $B$-module structure as follows: since $p$ is surjective, any $b\in B$ has a lift to a $x\in E$, so $b\cdot m:=x\cdot m$ for $m\in I$. Since any lift differs by an element $k\in I$ in the kernel, and
    $(x+k)\cdot m=x\cdot m+k\cdot m=x\cdot m$
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
   ${\begin{matrix}{\text{Spec}}\left({\frac {k[x,y]}{(y^{2}-x^{3})}}\right)&\to &{\text{Spec}}\left({\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\right)\\\downarrow &&\downarrow \\{\text{Spec}}(k)&\to &{\text{Spec}}(k[\varepsilon ])\end{matrix}}$
has the associated square-zero extension
   $0\to (\varepsilon )\to {\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\to {\frac {k[x,y]}{(y^{2}-x^{3})}}\to 0$
of $k$-algebras.

#### From more general deformations

But, because the idea of square zero-extensions is more general, deformations over $k[\varepsilon _{1},\varepsilon _{2}]$ where $\varepsilon _{1}\cdot \varepsilon _{2}=0$ will give examples of square-zero extensions.

#### Trivial square-zero extension

For a $B$-module $M$, there is a trivial square-zero extension given by $B\oplus M$ where the product structure is given by
    $(b,m)\cdot (b',m')=(bb',bm'+b'm)$
hence the associated square-zero extension is
    $0\to M\to B\oplus M\to B\to 0$
where the surjection is the projection map forgetting $M$.

## Construction

The general abstract construction of Exal follows from first defining a category of extensions ${\underline {\text{Exal}}}$ over a topos $T$ (or just the category of commutative rings), then extracting a subcategory where a base ring $A$ ${\underline {\text{Exal}}}_{A}$ is fixed, and then using a functor $\pi :{\underline {\text{Exal}}}_{A}(B,-)\to {\text{B-Mod}}$ to get the module of commutative algebra extensions ${\text{Exal}}_{A}(B,M)$ for a fixed $M\in {\text{Ob}}({\text{B-Mod}})$.

### General Exal

For this fixed topos, let ${\underline {\text{Exal}}}$ be the category of pairs $(A,p:E\to B)$ where $p:E\to B$ is a surjective morphism of $A$-algebras such that the kernel $I$ is square-zero, where morphisms are defined as commutative diagrams between $(A,p:E\to B)\to (A',p':E'\to B')$. There is a functor
    $\pi :{\underline {\text{Exal}}}\to {\text{Algmod}}$
sending a pair $(A,p:E\to B)$ to a pair $(A\to B,I)$ where $I$ is a $B$-module.

### Exal$_{A,}$ Exal$_A$(B, –)

Then, there is an overcategory denoted ${\underline {\text{Exal}}}_{A}$ (meaning there is a functor ${\underline {\text{Exal}}}_{A}\to {\displaystyle {\underline {\text{Exal}}}}$) where the objects are pairs $(A,p:E\to B)$, but the first ring $A$ is fixed, so morphisms are of the form
    $(A,p:E\to B)\to (A,p':E'\to B')$
There is a further reduction to another overcategory ${\underline {\text{Exal}}}_{A}(B,-)$ where morphisms are of the form
    $(A,p:E\to B)\to (A,p':E'\to B)$

### Exal$_A$(B,I )

Finally, the category ${\underline {\text{Exal}}}_{A}(B,I)$ has a fixed kernel of the square-zero extensions. Note that in ${\text{Algmod}}$, for a fixed $A,B$, there is the subcategory $(A\to B,I)$ where $I$ is a $B$-module, so it is equivalent to ${\text{B-Mod}}$. Hence, the image of ${\underline {\text{Exal}}}_{A}(B,I)$ under the functor $\pi$ lives in ${\text{B-Mod}}$.
The isomorphism classes of objects has the structure of a $B$-module since ${\underline {\text{Exal}}}_{A}(B,I)$ is a Picard stack, so the category can be turned into a module ${\text{Exal}}_{A}(B,I)$.

## Structure of Exal$_A$(B, I )

There are a few results on the structure of ${\underline {\text{Exal}}}_{A}(B,I)$ and ${\text{Exal}}_{A}(B,I)$ which are useful.

### Automorphisms

The group of automorphisms of an object $X\in {\text{Ob}}({\underline {\text{Exal}}}_{A}(B,I))$ can be identified with the automorphisms of the trivial extension $B\oplus M$ (explicitly, we mean automorphisms $B\oplus M\to B\oplus M$ compatible with both the inclusion $M\to B\oplus M$ and projection $B\oplus M\to B$). These are classified by the derivations module ${\text{Der}}_{A}(B,M)$. Hence, the category ${\underline {\text{Exal}}}_{A}(B,I)$ 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 ${\underline {\text{Exal}}}_{A}(B,-)$ describing the extensions of $I\oplus J$, there is an isomorphism
   ${\underline {\text{Exal}}}_{A}(B,I\oplus J)\cong {\underline {\text{Exal}}}_{A}(B,I)\times {\underline {\text{Exal}}}_{A}(B,J)$
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 $\varepsilon _{1},\varepsilon _{2}$ where $\varepsilon _{1}^{2}=\varepsilon _{1}\varepsilon _{2}=\varepsilon _{2}^{2}=0$ gives the isomorphism
   ${\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1})\oplus (\varepsilon _{2}))\cong {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1}))\times {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{2}))$
where $I$ 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
   $H^{1}(X,T_{X})\times H^{1}(X,T_{X})$
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 $A\to B$ over a topos $T$ (note taking $T$ as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphism
   ${\text{Exal}}_{A}(B,M)\xrightarrow {\simeq } {\text{Ext}}_{B}^{1}(\mathbf {L} _{B/A},M)$$^{(theoremIII.1.2.3)}$
So, given a commutative square of ring morphisms
   ${\begin{matrix}A'&\to &B'\\\downarrow &&\downarrow \\A&\to &B\end{matrix}}$
over $T$ there is a square
   ${\begin{matrix}{\text{Exal}}_{A}(B,M)&\to &{\text{Ext}}_{B}^{1}(\mathbf {L} _{B/A},M)\\\downarrow &&\downarrow \\{\text{Exal}}_{A'}(B',M)&\to &{\text{Ext}}_{B'}^{1}(\mathbf {L} _{B'/A'},M)\end{matrix}}$
whose horizontal arrows are isomorphisms and $M$ has the structure of a $B'$-module from the ring morphism.