See page 176 of [CK99] for the full background of this content.
Consider the affine case first, which should be enough since this stuf should be local. So \(S\) is an affine scheme over \(\mathbb C\), \(\mathcal N\) is an $\mathcal S$-module (probably coherent) and then we form the trivial extension \(S_{\mathcal N} = \Spec (\Gamma(\mathcal O_S)\oplus \Gamma(\mathcal N))\). This gives an infinitesmal extesion \(S\hookrightarrow S_{\mathcal N}\).
Now let
\begin{align*} \mathcal F:(\mathbb C - Schemes)\to (Sets) \end{align*}be a “contravariant moduli functor”, i.e. some functor which sends a scheme \(S\), as above, to all geometric objects of a certain type over \(S\). An example would be the functor of points of \(S\) or maybe the functor which sends \(S\) to all geometric vector bundles of a fixed rank over \(S\).
The tangent functor of this \(\mathcal F\) is then denoted \(T\mathcal F\) and is actually a collection of functors. We get one functor \(T\mathcal F(\alpha)\) for EVERY \(\alpha \in \mathcal F(S)\):
\begin{align*} T\mathcal F(\alpha):(\mathcal O_S-modules)\to (Sets) \end{align*}which takes a \(\mathcal O_{S}\)-module \(\mathcal N\) to the set of all elements of \(\mathcal F(S_{\mathcal N})\) which restrict to \(\alpha\) under the natural restriction map \(\mathcal F(S_{\mathcal N}) \to \mathcal F(S)\). All the functors making up \(T\mathcal F\) satisfy a natural base change property. Finally, we define the tangent-obstruction complex of \(\mathcal F\) to be a two-term complex of functors
\begin{align*} T^{1}\mathcal F \to T^2\mathcal F \end{align*}where the arrow is the “zero natural transformation” (this always exists for functors between abelian categories, think about it), the functor \(T^1\mathcal F\) is just the tangent functor defined above, and \(T^2\mathcal F\) gives "a reasonable obstruction theory for \(\mathcal F\).
Let’s do an example in two levels of specificity.
Example
Let \(S\) be some scheme over \(\mathbb C\) and \(\mathcal N\) some coherent sheaf over \(S\).
- We’re gonna take \(\mathcal F\) to be the functor which takes \(S\) to isomorphism classes of curves over \(S\).
- An object \([\alpha] \in \mathcal F(S)\) is then a flat family \(\alpha:\mathcal C\to S\) of curves over \(S\).
- Given a coherent sheaf \(\mathcal N\), we can form the trivial extension \(S_{\mathcal N}\) of \(S\) by \(\mathcal N\). What do families over this extension look like?
- Well, given a family \(\alpha: C'\to S' \in \mathcal F(S')\), we can pull back via \(\iota:S\hookrightarrow S'\) to obtain a family over \(S\):
which is the “natural restriction map” \(\mathcal F(S')\to \mathcal F(S)\).
- If this pulled back family is the original family \(C\to S\) we started with, then we can think of \(C'\to S'\) as an infinitesmal extension of the family \(\alpha\). Isomorphism classes of these extensions form the set \(\mathcal {TF}(\alpha)\).
Notice that \(\mathcal{TF}\) implicitly depends on the scheme \(S\), since its domain category is \((\mathcal O_S-modules)\).