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)

$\begin{array}{rl}0\to & {\mathrm{Der}}_B(C,L)\to {\mathrm{Der}}_A(C,L)\to {\mathrm{Der}}_A(B,L)\to \\ & {\mathrm{Exalcomm}}_B(C,L)\to {\mathrm{Exalcomm}}_A(C,L)\to {\mathrm{Exalcomm}}_A(B,L)\end{array}$

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 $\underset{\_}{\text{Exal}}$ 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

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

has the associated square-zero extension

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

of

$k$

-algebras.

From more general deformations

But, because the idea of square zero-extensions is more general, deformations over $k[{\epsilon }_1,{\epsilon }_2]$ where ${\epsilon }_1\cdot {\epsilon }_2=0$ 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

$(b,m)\cdot (b^{\prime },m^{\prime })=(b{b}^{\prime },b{m}^{\prime }+b^{\prime }m)$

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 $\underset{\_}{\text{Exal}}$ over a topos $T$ (or just the category of commutative rings), then extracting a subcategory where a base ring $A$ ${\underset{\_}{\text{Exal}}}_A$ is fixed, and then using a functor $\pi :{\underset{\_}{\text{Exal}}}_A(B,-)\to \text{B-Mod}$ 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 $\underset{\_}{\text{Exal}}$ 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 $(A,p:E\to B)\to (A^{\prime },p^{\prime }:E^{\prime }\to B^{\prime })$. There is a functor

$\pi :\underset{\_}{\text{Exal}}\to \text{Algmod}$

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

$(A,p:E\to B)\to (A,p^{\prime }:E^{\prime }\to B^{\prime })$

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

$(A,p:E\to B)\to (A,p^{\prime }:E^{\prime }\to B)$

Exal_A(B,I )

Finally, the category ${\underset{\_}{\text{Exal}}}_A(B,I)$ 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 ${\underset{\_}{\text{Exal}}}_A(B,I)$ under the functor $\pi$ lives in $\text{B-Mod}$. The isomorphism classes of objects has the structure of a $B$-module since ${\underset{\_}{\text{Exal}}}_A(B,I)$ 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 ${\underset{\_}{\text{Exal}}}_A(B,I)$ and ${\text{Exal}}_A(B,I)$ which are useful.

Automorphisms

The group of automorphisms of an object $X\in \text{Ob}({\underset{\_}{\text{Exal}}}_A(B,I))$ 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 ${\underset{\_}{\text{Exal}}}_A(B,I)$ 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

${\underset{\_}{\text{Exal}}}_A(B,-)$

describing the extensions of

$I\oplus J$

, there is an isomorphism

${\underset{\_}{\text{Exal}}}_A(B,I\oplus J)\cong {\underset{\_}{\text{Exal}}}_A(B,I)\times {\underset{\_}{\text{Exal}}}_A(B,J)$

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

${\epsilon }_1,{\epsilon }_2$

where

${\epsilon }_1^2={\epsilon }_1{\epsilon }_2={\epsilon }_2^2=0$

gives the isomorphism

${\underset{\_}{\text{Exal}}}_A(B,({\epsilon }_1)\oplus ({\epsilon }_2))\cong {\underset{\_}{\text{Exal}}}_A(B,({\epsilon }_1))\times {\underset{\_}{\text{Exal}}}_A(B,({\epsilon }_2))$

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

${\text{Exal}}_A(B,M)\stackrel{\simeq }{\to }{\text{Ext}}_B^1({\mathbf{L}}_{B/A},M)$

^{(theorem III.1.2.3)}

So, given a commutative square of ring morphisms

$\begin{array}{ccc}{A}^{\prime }& \to & B^{\prime }\\ \downarrow & & \downarrow \\ A& \to & B\end{array}$

over

$T$

there is a square

$\begin{array}{ccc}{\text{Exal}}_A(B,M)& \to & {\text{Ext}}_B^1({\mathbf{L}}_{B/A},M)\\ \downarrow & & \downarrow \\ {\text{Exal}}_{A^{\prime }}(B^{\prime },M)& \to & {\text{Ext}}_{B^{\prime }}^1({\mathbf{L}}_{B^{\prime }/A^{\prime }},M)\end{array}$

whose horizontal arrows are isomorphisms and

$M$

has the structure of a

$B^{\prime }$

-module from the ring morphism.

See also

• Deformation theory
• Cotangent complex
• Picard stack

References

• 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