Let \(A\) be noetherian, let \(X\) be projective over \(A\) with \(\OO_X(1)\) very ample, and let \(\mcf\) be coherent. Then there is \(n_0\) such that \[ H^i(X, \mcf(n)) = 0 \quad \text{for all } i > 0 \text{ and all } n \geq n_0 . \] [@Har10a, Theorem III.5.2]
(Dependence of the bound on the sheaf)
For a field \(k\) and an integer \(m\ge0\), the sheaf \(\OO_{\PP_k^1}(-m)\) has vanishing positive-degree cohomology after twisting by every \(n\ge m\). At \(n=m-2\), its first cohomology has dimension one [@Har10a, Theorem III.5.1]. Thus this family on \(\PP_k^1\) has no common twist bound as \(m\) varies; T-MODSERRE gives the explicit cohomology and global-generation bounds.
(Euler characteristics over a field)
Suppose \(A=k\) is a field. The dimensions \(h^i(X,\mcf(n))=\dim_k H^i(X,\mcf(n))\) are finite by T-COHFIN. For all sufficiently large \(n\), Serre vanishing gives \[ \chi(X,\mcf(n))=\sum_i(-1)^i h^i(X,\mcf(n))=h^0(X,\mcf(n)). \] If \(S\) is a finitely generated graded \(k\)-algebra with \(S_0=k\), generated in degree one, and \(M\) is a finite graded \(S\)-module with \(X=\Proj S\) and \(\mcf=\widetilde M\), then P-AGH259GAMMASTAR identifies the last term with \(\dim_k M_n\) for all sufficiently large \(n\). Consequently this Euler characteristic agrees in large degrees with the Hilbert polynomial of \(M\) [@Har10a, Chapter I, §7].
(Cohomological criterion for ampleness)
Let \(A\) be noetherian, let \(Y\) be proper over \(A\), and let \(\mcl\) be an invertible sheaf on \(Y\). Then \(\mcl\) is ample if and only if, for every coherent sheaf \(\mcg\) on \(Y\), there exists an integer \(N\) such that \[ H^i(Y,\mcg\otimes_{\OO_Y}\mcl^{\otimes n})=0 \quad\text{for every }i>0\text{ and every }n\ge N. \] [@Har10a, III.5.3]
Dependencies
T-COHFINFiniteness of cohomology for coherent sheaves on a projective schemeuses
Backlinks
D-COHEULERThe Euler characteristic and the Hilbert polynomialuses