Exalcomm

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

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

${\displaystyle (x+k)\cdot m=x\cdot m+k\cdot m=x\cdot m}$

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

${\displaystyle {\begin{matrix}{\text{Spec}}\left({\frac {k[x,y]}{(y^{2}-x^{3})}}\right)&\to &{\text{Spec}}\left({\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\right)\\\downarrow &&\downarrow \\{\text{Spec}}(k)&\to &{\text{Spec}}(k[\varepsilon ])\end{matrix}}}$

has the associated square-zero extension

${\displaystyle 0\to (\varepsilon )\to {\frac {k[x,y][\varepsilon ]}{(y^{2}-x^{3}+\varepsilon )}}\to {\frac {k[x,y]}{(y^{2}-x^{3})}}\to 0}$

of ${\displaystyle k}$-algebras.

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

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

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

hence the associated square-zero extension is

${\displaystyle 0\to M\to B\oplus M\to B\to 0}$

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

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

${\displaystyle \pi$ :{\underline {\text{Exal}}}\to {\text{Algmod}}}

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

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

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

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

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

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

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

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

${\displaystyle {\underline {\text{Exal}}}_{A}(B,I\oplus J)\cong {\underline {\text{Exal}}}_{A}(B,I)\times {\underline {\text{Exal}}}_{A}(B,J)}$

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 ${\displaystyle \varepsilon _{1},\varepsilon _{2}}$ where ${\displaystyle \varepsilon _{1}^{2}=\varepsilon _{1}\varepsilon _{2}=\varepsilon _{2}^{2}=0}$ gives the isomorphism

${\displaystyle {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1})\oplus (\varepsilon _{2}))\cong {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{1}))\times {\underline {\text{Exal}}}_{A}(B,(\varepsilon _{2}))}$

where ${\displaystyle I}$ 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

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

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