Show that the result of (11.2) is false without the projective hypothesis. For example, let \(X = \AA_k^n\), let \(P = (0, \ldots, 0)\), let \(U = X - P\), and let \(f: U \to X\) be the inclusion. Then the fibres of \(f\) all have dimension \(0\), but \[ R^{n-1} f_* \mco_U \neq 0 .\]
Solution
Write \[ A=k[x_1,\ldots,x_n], \qquad X=\Spec A, \qquad P=V(x_1,\ldots,x_n). \]
1. Every nonempty fibre of \(f\) is a single point, and the fibre over \(P\) is empty.
The map \(f:U\hookrightarrow X\) is an open immersion. Hence for \(x\in U\), \[ U\times_X\Spec\kappa(x)\cong\Spec\kappa(x), \] whereas \[ U\times_X\Spec\kappa(P)=\varnothing. \] Thus every fibre has dimension at most \(0\); in particular the morphism has the fibre-dimension bound appearing in the result whose projective hypothesis is being tested.
2. For every \(q\ge0\),\[ H^q(U,\mco_U) \cong \Gamma\bigl(X,R^qf_*\mco_U\bigr). \]
We have the finite affine cover \[ U=D(x_1)\cup\cdots\cup D(x_n), \] so the open immersion \(f\) is quasi-compact; every open immersion is quasi-separated. Since \(X\) is affine and \(\mco_U\) is quasicoherent, the affine-base higher-direct-image formula gives the displayed isomorphism. Equivalently, this is the collapse of the Leray spectral sequence after using quasi-coherence of higher direct images.
3. If \(n=1\), then\[ R^0f_*\mco_U\ne0. \]
In this case \[ U=D(x_1)=\Spec k[x_1,x_1^{-1}], \] so \[ H^0(U,\mco_U)=k[x_1,x_1^{-1}]\ne0. \] Step 2 with \(q=0\) gives \[ \Gamma(X,R^0f_*\mco_U)\ne0, \] hence the sheaf \(R^0f_*\mco_U\) is nonzero.
4. Assume \(n\ge2\). The cover\[ \mathcal U=\{D(x_i)\}_{i=1}^n \]computes \(H^\bullet(U,\mco_U)\) by its alternating Cech complex.
Every nonempty finite intersection is a distinguished affine open: \[ D(x_{i_0})\cap\cdots\cap D(x_{i_p}) = D(x_{i_0}\cdots x_{i_p}). \] Therefore the Cech-to-derived-cohomology map is an isomorphism for the quasicoherent sheaf \(\mco_U\); see the affine-intersection Cech comparison.
5. The top nonzero Cech cohomology group is\[ H^{n-1}(U,\mco_U) \cong \frac{A_{x_1\cdots x_n}} {\displaystyle\sum_{i=1}^n A_{x_1\cdots\widehat{x_i}\cdots x_n}}. \]
The degree-\(n-1\) term of the alternating Cech complex is the single localization \[ A_{x_1\cdots x_n}. \] There is no outgoing differential. The image of the degree-\(n-2\) differential is the sum of the images of the \(n\) localizations obtained by omitting one \(x_i\), with signs that do not affect the generated submodule. Taking the cokernel gives the displayed quotient.
6. The class\[ \left[\frac1{x_1\cdots x_n}\right] \in H^{n-1}(U,\mco_U) \]is nonzero.
The ring \[ A_{x_1\cdots x_n} = k[x_1^{\pm1},\ldots,x_n^{\pm1}] \] has the Laurent monomials \[ x_1^{a_1}\cdots x_n^{a_n}, \qquad (a_1,\ldots,a_n)\in\ZZ^n, \] as a \(k\)-basis.
For fixed \(i\), every Laurent monomial occurring in \[ A_{x_1\cdots\widehat{x_i}\cdots x_n} \] has exponent of \(x_i\) at least \(0\). Consequently the sum in the denominator of step 5 is spanned by Laurent monomials having at least one nonnegative exponent.
The monomial \[ x_1^{-1}\cdots x_n^{-1} \] has every exponent equal to \(-1\), so it does not lie in that sum. Its class is therefore nonzero.
7. For \(n\ge2\),\[ R^{n-1}f_*\mco_U\ne0. \]
By step 6, \[ H^{n-1}(U,\mco_U)\ne0. \] Step 2 identifies this group with \[ \Gamma\bigl(X,R^{n-1}f_*\mco_U\bigr). \] A sheaf with a nonzero global section is nonzero, proving the claim.
8. Q.E.D.
Step 1 gives the required fibre-dimension bound. Step 3 proves the assertion when \(n=1\), and steps <1>4–<1>7 prove it when \(n\ge2\). Thus the projective hypothesis in (11.2) cannot be omitted.