In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(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{align} 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{align}

where DerA(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 \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.

ExalA, ExalA(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)

ExalA(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 ExalA(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)[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}^1_B(\mathbf{L}_{B/A}, M) \\ \downarrow & & \downarrow \\ \text{Exal}_{A'}(B',M) & \to & \text{Ext}^1_{B'}(\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.

See also

References

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