Let \(k\) be an algebraically closed field and let \(X\) be a variety of dimension \(n\) over \(k\). Let \(\mce\) be a locally free sheaf of finite rank \(r>n\) on \(X\), and let \(V \subseteq \Gamma(X, \mce)\) be a vector space of global sections which generate \(\mce\). Then show that there is an element \(s \in V\) such that for each \(x \in X\) we have \(s_x \not\in \mfm_x \mce_x\). Conclude that there is a morphism \(\OO_X \to \mce\) giving rise to an exact sequence \[ 0 \to \OO_X \to \mce \to \mce' \to 0, \] where \(\mce'\) is also locally free.
The proof also works over any infinite field: every nonempty open subset of \(\AA_k^m\) has a \(k\)-point, since a nonzero polynomial cannot vanish on all of \(k^m\) when \(k\) is infinite. This follows by induction on \(m\) from the finiteness of the roots of a nonzero polynomial in one variable.
Hint
Use a method similar to the proof of Bertini’s theorem (II.8.18).
Solution
The value of a section at \(x\) means its image in the fibre \(\mce(x)=\mce_x\otimes_{\OO_{X,x}}\kappa(x)\). Thus the required condition is nonzero fibre value at every scheme point, not just nonzero germ.
1. There is a finite-dimensional subspace \(W\subseteq V\) which generates \(\mce\).
At every point choose finitely many sections from \(V\) that generate the stalk, using the assumed global generation and finite rank. Their images generate on a neighborhood: the cokernel of the map from a finite free sheaf is coherent and vanishes at that stalk, so it vanishes after shrinking. The variety \(X\) is quasi-compact, so finitely many such neighborhoods cover it. The span \(W\) of the sections chosen on that finite cover is finite-dimensional and generates at every point. Write \(m=\dim_kW\) and choose a basis \(s_1,\ldots,s_m\). In particular, \(m\ge r\).
2. Let \(I\subseteq X\times_k\AA_k^m\) be the incidence subscheme defined by the vanishing of the universal section \(\sum_j a_js_j\), where \(a_1,\ldots,a_m\) are the coordinates on \(\AA^m\).Then \(I\) is the total space of a vector bundle of rank \(m-r\) on \(X\), and\[ \dim I=n+m-r<m. \]
The evaluation map \(W\otimes_k\OO_X\twoheadrightarrow\mce\) is surjective. It splits locally because the target is locally free: on a framed open, choose local lifts of a basis. Its kernel is consequently locally free of rank \(m-r\).
On such an open, change the frame of \(W\otimes_k\OO_X\) according to this splitting. The vanishing of the universal section is then the vanishing of \(r\) independent fibre coordinates, leaving an affine space of dimension \(m-r\) over that open. These local descriptions identify \(I\) with the total space of the kernel bundle. In particular, \(I\) is integral and has dimension \(n+m-r\), since its trivializing opens are products of nonempty opens of the integral variety \(X\) with \(\AA^{m-r}\). The inequality follows from \(r>n\).
3. There is a section \(s\in W\) with no zero on \(X\).
Let \(Z\) be the reduced closure of the image of the second projection \(I\to\AA^m\). The induced dominant morphism from the integral variety \(I\) to \(Z\) gives an inclusion of function fields. Dimensions of integral finite-type schemes over a field equal the transcendence degrees of their function fields, so \[ \dim Z\le\dim I<m \] [@Har10a, Chapter I, §§1 and 4]. Thus \(\AA^m\setminus Z\) is a nonempty open subset and contains a \(k\)-rational point \(a=(a_1,\ldots,a_m)\), since \(k\) is algebraically closed. Choose this point and put \(s=\sum_j a_js_j\in W\subseteq V\).
The fibre of \(I\to\AA^m\) over \(a\) is exactly the zero subscheme of \(s\). It is empty because \(a\) is outside the closure of the image. Consequently \(s_x\notin\mathfrak m_x\mce_x\) for every \(x\in X\), as required. The argument uses the closure of the incidence image; it does not require that the projection be proper or that the image itself be closed.
4. The map \(\OO_X\to\mce\) defined by \(1\mapsto s\) is injective with a locally free cokernel of rank \(r-1\).
Fix \(x\in X\) and express \(s\) in a local frame \(e_1,\ldots,e_r\) as \(s=\sum_i b_ie_i\). Step 3 says that at least one \(b_i\) is outside the maximal ideal at \(x\). Reorder the frame so that this is \(b_1\), and shrink until \(b_1\) is a unit. The list \(s,e_2,\ldots,e_r\) is then a frame, since its change-of-basis determinant is \(b_1\). In this frame, the map from \(\OO_X\) is the inclusion of the first summand, with free cokernel of rank \(r-1\). These local descriptions prove global injectivity and local freeness of the cokernel \(\mce'\), giving the stated short exact sequence.
5. Q.E.D.
Steps <1>1–<1>3 produce the section in the prescribed space \(V\), and step 4 proves the exact sequence with locally free quotient.