Exalcomm Explained

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 .

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

\begin{align} 0 → &\operatorname{Der} _{B(C,L) →} \operatorname{Der} _{A(C,L) →} \operatorname{Der} _{A(B,L) →} \\ &\operatorname{Exalcomm} _{B(C,L) →} \operatorname{Exalcomm} _{A(C,L) →} \operatorname{Exalcomm} _{A(B,L) \end{align}} 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

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

\underline{ 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 ⋅ m : = x ⋅ m for m \in I . Since any lift differs by an element k \in I in the kernel, and(x + k) ⋅ m = x ⋅ m + k ⋅ m = x ⋅ 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} Spec \left( (k[x,y])/((y ^{2} - x ^{3} )) \right) & \to & Spec \left( (k[x,y][\varepsilon])/((y ^{2} - x ^{3} + \varepsilon)) \right) \\ \downarrow & & \downarrow \\ Spec (k) & \to & Spec (k[\varepsilon]) \end{matrix}

has the associated square-zero extension

0 \to (\varepsilon) \to (k[x,y][\varepsilon])/((y ^{2} - x ^{3} + \varepsilon)) \to (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 ⋅} \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 ⊕ M where the product structure is given by(b,m) ⋅ (b',m') = (bb',bm' + b'm) hence the associated square-zero extension is0 \to M \to B ⊕ 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{ Exal } over a topos T (or just the category of commutative rings), then extracting a subcategory where a base ring A \underline{ Exal }_A is fixed, and then using a functor \pi:\underline{ Exal }_A(B,-) \to \text to get the module of commutative algebra extensions Exal _{A(B,M)} for a fixed M \in Ob ( B-Mod ) .

General Exal

For this fixed topos, let

\underline{ 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{ Exal } \to \textsending 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{ Exal }_A (meaning there is a functor \underline{ Exal }_A \to ) 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{ 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{ Exal }_A(B,I) has a fixed kernel of the square-zero extensions. Note that in 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 B-Mod . Hence, the image of \underline{ Exal }_A(B,I) under the functor \pi lives in B-Mod .

The isomorphism classes of objects has the structure of a

B -module since \underline{ Exal }_A(B,I) is a Picard stack, so the category can be turned into a module Exal _{A(B,I)} .

Structure of Exal_A(B, I )

There are a few results on the structure of

\underline{ Exal }_A(B,I) and Exal _{A(B,I)} which are useful.

Automorphisms

The group of automorphisms of an object

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

\underline{ Exal }_A(B,I\oplus J) \cong \underline_A(B,I)\times \underline_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 ^{2} _{1} = \varepsilon _{1\varepsilon} _{2} = \varepsilon ^{2} _{2} = 0 gives the isomorphism

\underline{ Exal }_A(B,(\varepsilon_1) \oplus (\varepsilon_2)) \cong\underline_A(B,(\varepsilon_1))\times \underline_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) x} 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

Exal _{A(B,M)} \xrightarrow{\simeq} Ext ^{1 ( L} _{B/A} , M) ^{(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} Exal _{A(B,M)} & \to & Ext ^{1} _{B( L} _{B/A} , M) \\ \downarrow & & \downarrow \\ Exal _{A'} (B',M) & \to & Ext ^{1} _{B'} ( 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

• Deformation theory
• Cotangent complex
• Picard stack

References

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

• Tangent Spaces and Obstruction Theories - Olsson