- Source: Exalcomm
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
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.
= Definition
=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.
Remark
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.
= 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
Spec
(
k
[
x
,
y
]
(
y
2
−
x
3
)
)
→
Spec
(
k
[
x
,
y
]
[
ε
]
(
y
2
−
x
3
+
ε
)
)
↓
↓
Spec
(
k
)
→
Spec
(
k
[
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
)
→
k
[
x
,
y
]
[
ε
]
(
y
2
−
x
3
+
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.
From more general deformations
But, because the idea of square zero-extensions is more general, deformations over
k
[
ε
1
,
ε
2
]
{\displaystyle k[\varepsilon _{1},\varepsilon _{2}]}
where
ε
1
⋅
ε
2
=
0
{\displaystyle \varepsilon _{1}\cdot \varepsilon _{2}=0}
will give examples of square-zero extensions.
Trivial square-zero extension
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}
.
Construction
The general abstract construction of Exal 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
:
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}})}
.
= General Exal
=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
:
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.
= ExalA, ExalA(B, –)
=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)}
= ExalA(B,I )
=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
{\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)}
.
Structure of ExalA(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.
= Automorphisms
=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.
= Composition of extensions
=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.
Application
For example, the deformations given by infinitesimals
ε
1
,
ε
2
{\displaystyle \varepsilon _{1},\varepsilon _{2}}
where
ε
1
2
=
ε
1
ε
2
=
ε
2
2
=
0
{\displaystyle \varepsilon _{1}^{2}=\varepsilon _{1}\varepsilon _{2}=\varepsilon _{2}^{2}=0}
gives the isomorphism
Exal
_
A
(
B
,
(
ε
1
)
⊕
(
ε
2
)
)
≅
Exal
_
A
(
B
,
(
ε
1
)
)
×
Exal
_
A
(
B
,
(
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.
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
→
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)}
(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.
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