# Exalcomm

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Exalcomm
> Markdown URL: https://mediated.wiki/source/Exalcomm.md
> Source: https://en.wikipedia.org/wiki/Exalcomm
> Source revision: 1353496295
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{inline citations|date=May 2024}}

In [algebra](/source/algebra), '''Exalcomm''' is a [functor](/source/functor) classifying the extensions of a [commutative algebra](/source/commutative_algebra_(structure)) by a [module](/source/module_(mathematics)). More precisely, the elements of Exalcomm<sub>''k''</sub>(''R'',''M'') are [isomorphism class](/source/isomorphism_class)es of commutative [''k''-algebras](/source/algebra_over_a_field) ''E'' with a [homomorphism](/source/homomorphism) [onto](/source/surjective) the ''k''-algebra ''R'' whose [kernel](/source/kernel_(algebra)) 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 ring](/source/non-commutative_ring)s and algebras, and functors '''Exaltop''', '''Exantop''', and '''Exalcotop''' that take a [topology](/source/topological_space) into account.

"Exalcomm" is an abbreviation for "COMMutative ALgebra EXtension" (or rather for the corresponding French phrase). It was introduced by {{harvtxt|Grothendieck|Dieudonné|1964|loc=18.4.2}}.

Exalcomm is one of the [André–Quillen cohomology](/source/Andr%C3%A9%E2%80%93Quillen_cohomology) [group](/source/group_(mathematics))s and one of the [Lichtenbaum–Schlessinger functor](/source/Lichtenbaum%E2%80%93Schlessinger_functor)s.

Given homomorphisms of [commutative ring](/source/commutative_ring)s ''A''&nbsp;→&nbsp;''B''&nbsp;→&nbsp;''C'' and a ''C''-module ''L'' there is an [exact sequence](/source/exact_sequence) of ''A''-modules {{harv|Grothendieck|Dieudonné|1964|loc=20.2.3.1}}
:<math>\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}</math>
where Der<sub>''A''</sub>(''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](/source/Andr%C3%A9%E2%80%93Quillen_cohomology).

== Square-zero extensions ==
In order to understand the construction of Exal, the notion of square-zero extensions must be defined. Fix a [topos](/source/topos) <math>T</math> 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](/source/category_(mathematics)) <math>\underline{\text{Exal}}</math> we need to define what a square-zero extension actually is. Given a [surjective](/source/surjective) morphism of <math>A</math>-algebras <math>p: E \to B</math> it is called a '''square-zero extension''' if the kernel <math>I</math> of <math>p</math> has the property <math>I^2 = (0)</math> is the zero [ideal](/source/ideal_(ring_theory)).

==== Remark ====
Note that the kernel can be equipped with a <math>B</math>-module structure as follows: since <math>p</math> is surjective, any <math>b \in B</math> has a lift to a <math>x\in E</math>, so <math>b \cdot m := x\cdot m</math> for <math>m \in I</math>. Since any lift differs by an element <math>k \in I</math> in the kernel, and
:<math>(x + k)\cdot m = x\cdot m + k\cdot m = x\cdot m</math>
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 number](/source/dual_number)s. For example, a deformation over the dual numbers<blockquote><math>\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}</math></blockquote>has the associated square-zero extension<blockquote><math>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</math></blockquote>of <math>k</math>-algebras.

==== From more general deformations ====
But, because the idea of square zero-extensions is more general, deformations over <math>k[\varepsilon_1,\varepsilon_2]</math> where <math>\varepsilon_1\cdot \varepsilon_2 = 0</math> will give examples of square-zero extensions.

==== Trivial square-zero extension ====
For a <math>B</math>-module <math>M</math>, there is a trivial square-zero extension given by <math>B \oplus M</math> where the product structure is given by
:<math>(b,m)\cdot (b',m') = (bb',bm' + b'm)</math>
hence the associated square-zero extension is
:<math>0 \to M \to B\oplus M \to B \to 0</math>
where the surjection is the projection map forgetting <math>M</math>.

== Construction ==
The general abstract construction of Exal<ref name=":0">{{Cite book|last=Illusie|first=Luc|author-link=Luc Illusie|url=https://www.springer.com/gp/book/9783540056867|title=Complexe Cotangent et Deformations I|pages=151–168}}</ref> follows from first defining a category of extensions <math>\underline{\text{Exal}}</math> over a topos <math>T</math> (or just the category of commutative rings), then extracting a [subcategory](/source/subcategory) where a base ring <math>A</math> <math>\underline{\text{Exal}}_A</math> is fixed, and then using a functor <math>\pi:\underline{\text{Exal}}_A(B,-) \to \text{B-Mod}</math> to get the module of commutative algebra extensions <math>\text{Exal}_A(B,M)</math> for a fixed <math>M \in \text{Ob}(\text{B-Mod})</math>.

=== General Exal ===
For this fixed topos, let <math>\underline{\text{Exal}}</math> be the category of pairs <math>(A, p:E \to B)</math> where <math>p:E\to B</math> is a surjective morphism of <math>A</math>-algebras such that the kernel <math>I</math> is square-zero, where morphisms are defined as [commutative diagram](/source/commutative_diagram)s between <math>(A, p:E \to B) \to (A', p':E' \to B')</math>. There is a functor
:<math>\pi: \underline{\text{Exal}} \to \text{Algmod}</math>
sending a pair <math>(A, p:E \to B)</math> to a pair <math>(A\to B, I)</math> where <math>I</math> is a <math>B</math>-module.

=== Exal<sub>''A'',</sub> Exal<sub>''A''</sub>(''B'', –) ===
Then, there is an overcategory denoted <math>\underline{\text{Exal}}_A</math> (meaning there is a functor <math>\underline{\text{Exal}}_A \to {\displaystyle {\underline {\text{Exal}}}}</math>) where the objects are pairs <math>(A, p:E \to B)</math>, but the first ring <math>A</math> is fixed, so morphisms are of the form
:<math>(A, p:E \to B) \to (A, p':E' \to B')</math>
There is a further reduction to another overcategory <math>\underline{\text{Exal}}_A(B,-)</math> where morphisms are of the form
:<math>(A, p:E \to B) \to (A, p':E' \to B)</math>

=== Exal<sub>''A''</sub>(''B'',''I'') ===
Finally, the category <math>\underline{\text{Exal}}_A(B,I)</math> has a fixed kernel of the square-zero extensions. Note that in <math>\text{Algmod}</math>, for a fixed <math>A,B</math>, there is the subcategory <math>(A\to B, I)</math> where <math>I</math> is a <math>B</math>-module, so it is equivalent to <math>\text{B-Mod}</math>. Hence, the image of <math>\underline{\text{Exal}}_A(B,I)</math> under the functor <math>\pi</math> lives in <math>\text{B-Mod}</math>.

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

== Structure of Exal<sub>''A''</sub>(''B'', ''I'') ==
There are a few results on the structure of <math>\underline{\text{Exal}}_A(B,I)</math> and <math>\text{Exal}_A(B,I)</math> which are useful.

=== Automorphisms ===
The [group of automorphisms](/source/automorphism_group) of an object <math>X \in \text{Ob}(\underline{\text{Exal}}_A(B,I)
)</math> can be identified with the [automorphism](/source/automorphism)s of the trivial extension <math>B\oplus M</math> (explicitly, we mean automorphisms <math>B\oplus M \to B\oplus M</math> compatible with both the inclusion <math>M\to B \oplus M</math> and projection <math>B\oplus M \to B</math>). These are classified by the derivations module <math>\text{Der}_A(B,M)</math>. Hence, the category <math>\underline{\text{Exal}}_A(B,I)</math> is a torsor. In fact, this could also be interpreted as a [Gerbe](/source/Gerbe) since this is a group acting on a stack.

=== Composition of extensions ===
There is another useful result about the categories <math>\underline{\text{Exal}}_A(B,-)</math> describing the extensions of <math>I\oplus J</math>, there is an [isomorphism](/source/isomorphism)<blockquote><math>\underline{\text{Exal}}_A(B,I\oplus J) \cong \underline{\text{Exal}}_A(B,I)\times \underline{\text{Exal}}_A(B,J)</math></blockquote>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 <math>\varepsilon_1,\varepsilon_2</math> where <math>\varepsilon_1^2 = \varepsilon_1\varepsilon_2 = \varepsilon_2^2 = 0</math> gives the isomorphism<blockquote><math>\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))</math></blockquote>where <math>I</math> is the module of these two infinitesimals. In particular, when relating this to Kodaira-Spencer theory, and using the comparison with the [cotangent complex](/source/cotangent_complex) (given below) this means all such deformations are classified by<blockquote><math>H^1(X,T_X)\times H^1(X,T_X)</math></blockquote>hence they are just a pair of first order deformations paired together.

== Relation with the cotangent complex ==
The [cotangent complex](/source/cotangent_complex) contains all of the information about a deformation problem, and it is a fundamental theorem that given a morphism of rings <math>A \to B</math> over a topos <math>T</math> (note taking <math>T</math> as the point topos shows this generalizes the construction for general rings), there is a functorial isomorphism<blockquote><math>\text{Exal}_A(B,M) \xrightarrow{\simeq} \text{Ext}_B^1(\mathbf{L}_{B/A}, M)</math><ref name=":0" /><sup>(theorem III.1.2.3)</sup></blockquote>So, given a commutative square of ring morphisms<blockquote><math>\begin{matrix}
A' & \to & B' \\
\downarrow & & \downarrow \\
A & \to & B
\end{matrix}</math></blockquote>over <math>T</math> there is a square<blockquote><math>\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}</math></blockquote>whose horizontal arrows are isomorphisms and <math>M</math> has the structure of a <math>B'</math>-module from the ring morphism.

== See also ==

* [Deformation theory](/source/Deformation_theory)
* [Cotangent complex](/source/Cotangent_complex)
* [Picard stack](/source/Picard_stack)

==References==
<references />

*[https://web.archive.org/web/20200429024215/https://math.berkeley.edu/~molsson/MSRISummer07.pdf Tangent Spaces and Obstruction Theories] - Olsson
*{{EGA|book=4-1| pages = 65}}
*{{Citation | last1=Weibel | first1=Charles A. |author-link=Charles Weibel | title=An introduction to homological algebra | url=https://books.google.com/books?id=flm-dBXfZ_gC | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-43500-0 | id={{ISBN|978-0-521-55987-4}}, {{MR|1269324}} | year=1994 | volume=38}}

Category:Homological algebra

---
Adapted from the Wikipedia article [Exalcomm](https://en.wikipedia.org/wiki/Exalcomm) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Exalcomm?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.