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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)

0 → → Der B ⁡ ⁡ ( C , L ) → → Der A ⁡ ⁡ ( C , L ) → → Der A ⁡ ⁡ ( B , L ) → → Exalcomm B ⁡ ⁡ ( C , L ) → → Exalcomm A ⁡ ⁡ ( C , L ) → → Exalcomm A ⁡ ⁡ ( B , L ) {\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 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

• Definition

• Examples

• Construction

• General Exal

• Exal A , Exal A ( B , –)

• Exal A ( B , I )

• Structure of Exal A ( B , I )

• Automorphisms

• Composition of extensions

• Relation with the cotangent complex

• See also

• References

In order to understand the construction of Exal, the notion of square-zero extensions must be defined. Fix a topos T {\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.

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

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

( x + k ) ⋅ ⋅ m = x ⋅ ⋅ m + k ⋅ ⋅ m = x ⋅ ⋅ m {\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.

Square-zero extensions are a generalization of deformations over the dual numbers. For example, a deformation over the dual numbers

Spec ( k [ x , y ] ( y 2 − − x 3 ) ) → → Spec ( k [ x , y ] [ ε ε ] ( y 2 − − x 3 + ε ε ) ) ↓ ↓ ↓ ↓ Spec ( k ) → → Spec ( k [ ε ε ] ) {\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

0 → → ( ε ε ) → → k [ x , y ] [ ε ε ] ( y 2 − − x 3 + ε ε ) → → k [ x , y ] ( y 2 − − x 3 ) → → 0 {\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 k {\displaystyle k} -algebras.

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

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

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

hence the associated square-zero extension is

0 → → M → → B ⊕ ⊕ M → → B → → 0 {\displaystyle 0\to M\to B\oplus M\to B\to 0}

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

The general abstract construction of Exalcite-ref-0-1-0[1] follows from first defining a category of extensions Exal _ _ {\displaystyle {\underline {\text{Exal}}}} over a topos T {\displaystyle T} (or just the category of commutative rings), then extracting a subcategory where a base ring A {\displaystyle A} Exal _ _ A {\displaystyle {\underline {\text{Exal}}}_{A}} is fixed, and then using a functor π π : Exal _ _ A ( B , − − ) → → B-Mod {\displaystyle \pi :{\underline {\text{Exal}}}_{A}(B,-)\to {\text{B-Mod}}} to get the module of commutative algebra extensions Exal A ( B , M ) {\displaystyle {\text{Exal}}_{A}(B,M)} for a fixed M ∈ ∈ Ob ( B-Mod ) {\displaystyle M\in {\text{Ob}}({\text{B-Mod}})} .

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

π π : Exal _ _ → → Algmod {\displaystyle \pi :{\underline {\text{Exal}}}\to {\text{Algmod}}}

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

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

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

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

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

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

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

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

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

There is another useful result about the categories Exal _ _ A ( B , − − ) {\displaystyle {\underline {\text{Exal}}}_{A}(B,-)} describing the extensions of I ⊕ ⊕ J {\displaystyle I\oplus J} , there is an isomorphism

Exal _ _ A ( B , I ⊕ ⊕ J ) ≅ ≅ Exal _ _ A ( B , I ) × × Exal _ _ A ( B , J ) {\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.

For example, the deformations given by infinitesimals ε ε 1 , ε ε 2 {\displaystyle \varepsilon _{1},\varepsilon _{2}} where ε ε 1 2 = ε ε 1 ε ε 2 = ε ε 2 2 = 0 {\displaystyle \varepsilon _{1}^{2}=\varepsilon _{1}\varepsilon _{2}=\varepsilon _{2}^{2}=0} gives the isomorphism

Exal _ _ A ( B , ( ε ε 1 ) ⊕ ⊕ ( ε ε 2 ) ) ≅ ≅ Exal _ _ A ( B , ( ε ε 1 ) ) × × Exal _ _ A ( B , ( ε ε 2 ) ) {\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 I {\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

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

hence they are just a pair of first order deformations paired together.

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

Exal A ( B , M ) → ≃ ≃ Ext B 1 ( L B / A , M ) {\displaystyle {\text{Exal}}_{A}(B,M)\xrightarrow {\simeq } {\text{Ext}}_{B}^{1}(\mathbf {L} _{B/A},M)} cite-ref-0-1-1[1](theorem III.1.2.3)

So, given a commutative square of ring morphisms

A ′ → → B ′ ↓ ↓ ↓ ↓ A → → B {\displaystyle {\begin{matrix}A'&\to &B'\\\downarrow &&\downarrow \\A&\to &B\end{matrix}}}

over T {\displaystyle T} there is a square

Exal A ( B , M ) → → Ext B 1 ( L B / A , M ) ↓ ↓ ↓ ↓ Exal A ′ ( B ′ , M ) → → Ext B ′ 1 ( L B ′ / A ′ , M ) {\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 M {\displaystyle M} has the structure of a B ′ {\displaystyle B'} -module from the ring morphism.

• Deformation theory

• Cotangent complex

• Picard stack

cite-note-0-11. ↑ citerefillusieIllusie, Luc. Complexe Cotangent et Deformations I. pp. 151–168.

• Tangent Spaces and Obstruction Theories - Olsson

• citerefgrothendieckdieudonn-1964Grothendieck, 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.

• citerefweibel1994Weibel, 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