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^[1] 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 ${\displaystyle \pi$ 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

{\displaystyle \pi

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 contangent 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

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

References

1 2 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, MR 1269324

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[:0-1] 1 2 Illusie, Luc. Complexe Cotangent et Deformations I. pp. 151–168.

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