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)

${\displaystyle {\begin{aligned}0\rightarrow \;&\operatorname {Der} _{B}(C,L)\rightarrow \operatorname {Der} _{A}(C,L)\rightarrow \operatorname {Der} _{A}(B,L)\rightarrow \\&\operatorname {Exalcomm} _{B}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(B,L)\end{aligned}}}$

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

Remark

Note that the kernel can be equipped with a ${\displaystyle B}$-module structure as follows: since ${\displaystyle p}$ is surjective, any ${\displaystyle b\in B}$ has a lift to a ${\displaystyle x\in E}$, so ${\displaystyle b\cdot m:=x\cdot m}$ for ${\displaystyle m\in I}$. Since any lift differs by an element ${\displaystyle k\in I}$ in the kernel, and

${\displaystyle (x+k)\cdot m=x\cdot m+k\cdot m=x\cdot m}$

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

${\displaystyle {\begin{matrix}{\text{Spec}}\left({\frac {k[x,y]}{(y^{2}-x^{3})}}\right)&\to &{\text{Spec}}\left({\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\right)\\\downarrow &&\downarrow \\{\text{Spec}}(k)&\to &{\text{Spec}}(k[\varepsilon ])\end{matrix}}}$

has the associated square-zero extension

${\displaystyle 0\to (\varepsilon )\to {\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\to {\frac {k[x,y]}{(y^{2}-x^{3})}}\to 0}$

of ${\displaystyle k}$-algebras.

From more general deformations

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

Trivial square-zero extension

For a ${\displaystyle B}$-module ${\displaystyle M}$, there is a trivial square-zero extension given by ${\displaystyle B\oplus M}$ where the product structure is given by

${\displaystyle (b,m)\cdot (b',m')=(bb',bm'+b'm)}$

hence the associated square-zero extension is

${\displaystyle 0\to M\to B\oplus M\to B\to 0}$

where the surjection is the projection map forgetting ${\displaystyle M}$.

Construction

The general abstract construction of Exal^[1] follows from first defining a category of extensions ${\displaystyle {\underline {\text{Exal}}}}$ over a topos ${\displaystyle T}$ (or just the category of commutative rings), then extracting a subcategory where a base ring ${\displaystyle A}$ ${\displaystyle {\underline {\text{Exal}}}_{A}}$ is fixed, and then using a functor ${\displaystyle \pi$ :{\underline {\text{Exal}}}_{A}(B,-)\to {\text{B-Mod}}} to get the module of commutative algebra extensions ${\displaystyle {\text{Exal}}_{A}(B,M)}$ for a fixed ${\displaystyle M\in {\text{Ob}}({\text{B-Mod}})}$.

General Exal

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

${\displaystyle \pi$ :{\underline {\text{Exal}}}\to {\text{Algmod}}}

sending a pair ${\displaystyle (A,p:E\to B)}$ to a pair ${\displaystyle (A\to B,I)}$ where ${\displaystyle I}$ is a ${\displaystyle B}$-module.

Exal_A, Exal_A(B, –)

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

${\displaystyle (A,p:E\to B)\to (A,p':E'\to B')}$

There is a further reduction to another overcategory ${\displaystyle {\underline {\text{Exal}}}_{A}(B,-)}$ where morphisms are of the form

${\displaystyle (A,p:E\to B)\to (A,p':E'\to B)}$

Exal_A(B,I )

Finally, the category ${\displaystyle {\underline {\text{Exal}}}_{A}(B,I)}$ has a fixed kernel of the square-zero extensions. Note that in ${\displaystyle {\text{Algmod}}}$, for a fixed ${\displaystyle A,B}$, there is the subcategory ${\displaystyle (A\to B,I)}$ where ${\displaystyle I}$ is a ${\displaystyle B}$-module, so it is equivalent to ${\displaystyle {\text{B-Mod}}}$. Hence, the image of ${\displaystyle {\underline {\text{Exal}}}_{A}(B,I)}$ under the functor ${\displaystyle \pi }$ lives in ${\displaystyle {\text{B-Mod}}}$.

The isomorphism classes of objects has the structure of a ${\displaystyle B}$-module since ${\displaystyle {\underline {\text{Exal}}}_{A}(B,I)}$ is a Picard stack, so the category can be turned into a module ${\displaystyle {\text{Exal}}_{A}(B,I)}$.

Structure of ExalA(B, I )

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

Automorphisms

The group of automorphisms of an object ${\displaystyle X\in {\text{Ob}}({\underline {\text{Exal}}}_{A}(B,I))}$ can be identified with the automorphisms of the trivial extension ${\displaystyle B\oplus M}$ (explicitly, we mean automorphisms ${\displaystyle B\oplus M\to B\oplus M}$ compatible with both the inclusion ${\displaystyle M\to B\oplus M}$ and projection ${\displaystyle B\oplus M\to B}$). These are classified by the derivations module ${\displaystyle {\text{Der}}_{A}(B,M)}$. Hence, the category ${\displaystyle {\underline {\text{Exal}}}_{A}(B,I)}$ 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 ${\displaystyle {\underline {\text{Exal}}}_{A}(B,-)}$ describing the extensions of ${\displaystyle I\oplus J}$, there is an isomorphism

${\displaystyle {\underline {\text{Exal}}}_{A}(B,I\oplus J)\cong {\underline {\text{Exal}}}_{A}(B,I)\times {\underline {\text{Exal}}}_{A}(B,J)}$

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 ${\displaystyle \varepsilon _{1},\varepsilon _{2}}$ where ${\displaystyle \varepsilon _{1}^{2}=\varepsilon _{1}\varepsilon _{2}=\varepsilon _{2}^{2}=0}$ gives the isomorphism

${\displaystyle {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1})\oplus (\varepsilon _{2}))\cong {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1}))\times {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{2}))}$

where ${\displaystyle I}$ 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

${\displaystyle H^{1}(X,T_{X})\times H^{1}(X,T_{X})}$

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 ${\displaystyle A\to B}$ over a topos ${\displaystyle T}$ (note taking ${\displaystyle T}$ as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphism

${\displaystyle {\text{Exal}}_{A}(B,M)\xrightarrow {\simeq } {\text{Ext}}_{B}^{1}(\mathbf {L} _{B/A},M)}$^{[1](theorem III.1.2.3)}

So, given a commutative square of ring morphisms

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

over ${\displaystyle T}$ there is a square

${\displaystyle {\begin{matrix}{\text{Exal}}_{A}(B,M)&\to &{\text{Ext}}_{B}^{1}(\mathbf {L} _{B/A},M)\\\downarrow &&\downarrow \\{\text{Exal}}_{A'}(B',M)&\to &{\text{Ext}}_{B'}^{1}(\mathbf {L} _{B'/A'},M)\end{matrix}}}$

whose horizontal arrows are isomorphisms and ${\displaystyle M}$ has the structure of a ${\displaystyle B'}$-module from the ring morphism.

See also

• Deformation theory
• Cotangent complex
• Picard stack

References

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

• Tangent Spaces and Obstruction Theories - Olsson
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 65. doi:10.1007/bf02684747. MR 0173675.
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-43500-0, ISBN 978-0-521-55987-4, MR1269324