# Infinitesimal Extensions by Coherent Sheaves

Last time we looked at what the infinitesimal lifting property was and that a particularly nice class of schemes always satisfies it. A related concept is that of an infinitesimal extension of a scheme. Suppose ${X}$ is a scheme of finite type over ${k}$ and ${\mathcal{F}\in Coh(X)}$. A pair ${(X', \mathcal{I})}$ is an infinitesimal extension of ${X}$ by ${\mathcal{F}}$ if ${\mathcal{I}}$ is a sheaf of ideals satisfying ${\mathcal{I}^2=0}$ and as schemes ${(X', \mathcal{O}_X'/\mathcal{I})\simeq (X, \mathcal{O}_X)}$.

Given ${(X, \mathcal{F})}$, as an example we always have the trivial infinitesimal extension by ${\mathcal{F}}$, which is ${(X, \mathcal{F})}$ where the structure sheaf on ${X}$ is given by ${\mathcal{O}_X'=\mathcal{O}_X\oplus\mathcal{F}}$ where multiplication is defined by ${(a\oplus f)\cdot (a'\oplus f')=aa'\oplus (af'+a'f)}$. Now notice that the way that ${\mathcal{F}}$ is a sheaf of ideals given this new structure sheaf, is ${0\oplus \mathcal{F}\subset \mathcal{O}_X'}$. Take ${(0,f), (0,f')\in\mathcal{F}}$ and see what happens ${(0,f)\cdot (0,f')=(0,0)}$. Thus ${\mathcal{F}^2=0}$ and clearly ${\mathcal{O}_X'/\mathcal{F}\simeq \mathcal{O}_X}$, so this is really an infinitesimal extension.

Let’s go back to our nice simple case from last time: Let ${X=}$ Spec ${A}$ be nonsingular. Then the claim is that any infinitesimal extension by a coherent sheaf ${\mathcal{F}}$ is isomorphic to the trivial one.

This immediately converts to an algebra problem. Let ${M}$ be the ${A}$-module such that ${\mathcal{F}=\tilde{M}}$. Suppose we have an infinitesimal extension ${(Spec A', \mathcal{I})}$. Then ${I}$ is an ideal of ${A'}$ with ${I^2=0}$, ${A'/I\simeq A}$, and ${I\simeq M}$ as an ${A}$-module.

This gives us an exact sequence ${0\rightarrow I\rightarrow A'\rightarrow A\rightarrow 0}$. Consider the isomorphism ${f: A'/I\rightarrow A}$, by last time there exists a lift ${A'/I\rightarrow A'}$. This is a retraction and hence the sequence splits giving ${A'\simeq A\oplus I\simeq A\oplus M}$ (we do need to check that the multiplication is the right one, but it is by computation). Taking the sheaves associated to this, we see that ${\mathcal{O}_X'\simeq \mathcal{O}_X\oplus \mathcal{F}}$, the trivial extension.

It’s good to notice that by what we did yesterday that these lifts are not actually unique. They are actually “parametrized” in some sense by ${Hom_A(\Omega_{A/k}, M)\simeq H^0(X, \mathcal{F}\otimes \mathcal{T})}$ where ${\mathcal{T}}$ is the tangent sheaf. So it is reasonable to guess that ${\mathcal{F}\otimes \mathcal{T}}$ plays a role here. Since we’ve classified the infinitesimal extensions of non-singular affine varieties, and non-singular varieties are locally of that form, let’s extend the result.

If ${X}$ is a non-singular variety over ${k=\overline{k}}$, and ${\mathcal{F}\in Coh(X)}$, then there is a one-to-one correspondence between the set of infinitesimal extensions of ${\mathcal{F}}$ by ${X}$ up to isomorphism and ${H^1(X, \mathcal{F}\otimes \mathcal{T})}$.

Take a finite affine cover of ${X}$, ${\{U_i=Spec(A_i)\}_1^n}$. Then we have a natural iso ${\check{H}^p(\mathcal{U}, \mathcal{F}\otimes\mathcal{T})\simeq H^p(X, \mathcal{F}\otimes\mathcal{T})}$, so we’ll think in terms of ${\check{C}}$ech cocycles.

If ${(X', \mathcal{I})}$ is an infinitesimal extension, then on each affine it must look trivial, so ${\mathcal{I}\big|_{U_i}\simeq \tilde{N_i}\simeq \mathcal{O}_{U_i}\oplus \mathcal{F}\big|_{U_i}}$. So ${A_i\oplus M_i=H^0(U_i, \mathcal{O}_X\oplus \mathcal{F})}$. Any two liftings differ by a section of ${Hom(\Omega_{X/k}, \mathcal{F})\simeq \mathcal{F}\otimes \Omega_{X/k}^* = \mathcal{F}\otimes\mathcal{T}}$. This is exactly saying that the pairwise intersections satisfy the cocycle condition and hence gives us an element of ${H^1(X, \mathcal{F}\otimes \mathcal{T})}$.

This basically reverses for the converse. Given an element of ${H^1(X, \mathcal{F}\otimes\mathcal{T})}$, then it satisfies the cocycle condition and hence patches by differences in the right thing to give an infinitesimal extension.

1. What are $\mathcal{O}$ and $\Omega$ ?
Depending on what you know, this could be a rather hard thing to answer. I’ll just throw some words out there. $\mathcal{O}$ is the structure sheaf (sheaf of regular functions), and $\Omega$ is the sheaf of relative differentials.