Setup \(X\) is a scheme (or a stack, it won’t matter) and \(\varphi:B\to X\) is an affine bundle of rank \(n\) over \(X\). This means it is a \(F\)-torsor for some rank \(n\) vector bundle over \(X\). So our terms:
- \(\mathcal F\) a locally free sheaf of rank \(n\) on \(X\)
- \( F= \Spec \Sym \mathcal F^\vee\) is the total space of \(\mathcal F\)
- \( \varphi:B\to X\) is a \(F\)-torsor over \(X\)
Construction Set \(\mathcal A = \varphi_*\mathcal O_B\) and define \(\mathcal A^{\leq 1}\subseteq \mathcal A\) to be the subsheaf of affine-linear sections. That this is globally well-defined is a descent argument: \(\mathcal O_B\cong \mathcal O_X[y_1,...,y_n]\) locally, and because the transition functions are affine, they preserve \(\mathcal A^{\leq 1}\). Alternatively, for affine bundles one can observe that \(\Omega_{B/X} \cong \varphi^*\mathcal F^\vee\), since translations act trivially on relative differentials the affine maps act as if they were linear on differentials. Then \(f \in \mathcal A\) lies in \(\mathcal A^{\leq 1}\) if and only if \(d_{B/X}f \in \varphi_*\varphi^*\mathcal F^{\vee}\) lies in the image of the unit \(\mathcal F^\vee \to \varphi_*\varphi^*\mathcal F^\vee\), i.e. if its fiberwise differential is constant along fibers.
Anyways, there are two maps involving \(\mathcal A^{\leq 1}\);
- the constant part: \(\mathcal O_X\to \mathcal A^{\leq 1}\) which we just call \(1\), identifiying \(\mathcal O_X\) with the constant sections of \(\mathcal A^{\leq 1}\), and
- the linear part: \( \mathcal A^{\leq 1}\xrightarrow{d_{B/X}} \mathcal F\), which is locally defined \(d_{B/X}(a_1y_1 + ... + a_n y_n + b) = (a_1,...,a_n)\in \mathcal F\).
Really spelling this out: on an affine we have
- \(\mathcal O_X = R\)
- \(\mathcal O_F = R[y_1,...,y_n]\)
- \(\mathcal F = \{a_1y_1 + ... + a_ny_n \in R[y_1,...,y_n]^{\deg = 1}\} \cong R^{\oplus n}\)
- \(\mathcal A^{\leq 1} = \{a_1y + ... + a_ny_n + b \in R[y_1,...,y_n]^{\deg \leq 1}\} \cong R^{\oplus (n+1)}\)
This makes it obvious that \(\mathcal A^{\leq 1}\) is a locally free sheaf of rank \(n + 1\); call it \(\mathcal E\). These two maps fit into the exact sequence
\begin{align*} 0 \to \mathcal O_X\xrightarrow{1} \mathcal E \xrightarrow{d_{B/X}} \mathcal F\to 0. \end{align*}Dualizing gives us
\begin{align*} 0 \to \mathcal F^\vee \xrightarrow{d_{B/X}^\vee} \mathcal E^\vee \xrightarrow{\tau} \mathcal O_X\to 0 \end{align*}where \(\tau(\xi) = \xi(1)\) evaluates a section \(\xi\) at the constact function \(1\in \mathcal E = \mathcal A^{\leq 1}\) . Taking \(\Spec\Sym(-)\) gives us the exact sequence of vector bundles
\begin{align*} 0 \to F\to E\xrightarrow{\rho} \mathbb A^1_{X}\to 0. \end{align*}Geometry The inclusion \(\mathcal A^{\leq 1}\hookrightarrow \mathcal A\) induces a surjection \(\Sym(\mathcal A^{\leq 1})\twoheadrightarrow \mathcal A.\) It’s surjective because \(\mathcal A\) is locally generated in degree \(\leq 1\). This gives us an inclusion \(B\hookrightarrow \operatorname{Tot}(\mathcal A^{\leq 1}) = \operatorname{Tot}(\mathcal E) = E\). What precisely is the ideal? Locally, the surjection looks like
\begin{align*} R[T,Y_1,...,Y_n] \to R[Y_1,...,Y_n] \end{align*}with \(T\) the formal variable attached to \(1 \in \mathcal A^{\leq 1}\). Hence the map is \(T\mapsto 1, Y_i \mapsto Y_i\), so the ideal is \((T - 1)\). Globally, this section \(T\) is the fiberwise-linear function on \(E\) attached to \(1 \in \mathcal E^\vee\), the composite
\begin{align*} t:E\xrightarrow{\rho} X\times \mathbb A^1 \xrightarrow{\pr} \mathbb A^1, \end{align*}since \(\tau\) was defined as evaluation at \(1\). Thus \(t^{-1}(1) = B\).
We also have that \(t^{-1}(0) = F\), since functionals killing constants are precisely functionals on \(\mathcal A^{\leq 1}/\mathcal O\cong \mathcal F\). Thus, \(E\) is a family over \(\mathbb A^1\) whose central fiber is the trivial torsor \(F\) and whose fiber over \(1\) (and every other nonzero point) is \(B\).
Ext groups and Cohomology Notice that
\begin{align*} \Ext^1_X(\mathcal F, \mathcal O_X) \cong \Ext^1_X(\mathcal O_X, \mathcal F^\vee) \cong H^1_X(X, \mathcal F). \end{align*}This is the one line cohomological justification for the above construction, when \(D\) is any \(F\)-torsor it has a canonical associated extension
\begin{align*} 0\to\mathcal F\to \mathcal F'\to \mathcal O_X\to 0, \end{align*}and \(\Spec Sym(\mathcal F'^\vee)\) will give you the desired rank \(n+1\) vector bundle \(E\) into which \(D\) embeds. Also visible here: \(D\) is a split/trivial torsor exactly when the above extension class \([\mathcal F']\) is trivial i.e. when the above short exact sequence splits.