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.


3 thoughts on “Infinitesimal Extensions by Coherent Sheaves

  1. 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.

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