View source for Ample sheaf

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A generalization of the concept of an ample [[Invertible sheaf|invertible sheaf]]. Let $X$ be a Noetherian scheme over a field $k$, and let $\mathcal E$ be a locally free sheaf on $X$ (that is, the sheaf of sections of some algebraic vector bundle $E\to X$). The sheaf $\mathcal E$ is called ample if for each coherent sheaf $\mathcal F$ on $X$ there exists an integer $n_0$, depending on $\mathcal F$, such that the sheaf $\mathcal F\otimes S^n\mathcal E$ for $n\geq n_0$ is generated by its global sections (here $S^n\mathcal E$ denotes the $n$-th symmetric power of $\mathcal E$).

A locally free sheaf $\mathcal E$ on $X$ is ample if and only if the invertible tautological sheaf $\mathcal L(\mathcal E)$ on the projectivization $P(E)$ of the bundle $E$ is ample. Another criterion of ampleness is that for each coherent sheaf $\mathcal F$ on $X$ there must exist an integer $n_0$, depending on $\mathcal F$, such that the cohomology group $H^i(X,\mathcal F\otimes S^n\mathcal E)$ is zero for $n\geq n_0$ and $i>0$. If the sheaves $\mathcal E$ and $\mathcal F$ are ample then $\mathcal E\otimes\mathcal F$ is an ample sheaf [[#References|[1]]]. If $X$ is a non-singular projective curve, then a sheaf $\mathcal E$ on $X$ is ample if and only if $\mathcal E$ and all its quotient sheaves have positive degree [[#References|[2]]]. The tangent sheaf on $P^N$ is ample for any $N$ (see [[#References|[1]]]). The converse also holds: Any non-singular $N$-dimensional algebraic variety with an ample tangent sheaf is isomorphic to $P^N$ (see [[#References|[1]]], [[#References|[3]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Ample vector bundles"  ''Publ. Math. IHES'' , '''29'''  (1966)  pp. 319–350</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Ample vector bundles on curves"  ''Nagoya Math. J.'' , '''43'''  (1971)  pp. 73–89</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> M. Demazure, "Caractérisations de l'espace projectif (conjectures de Hartshorne et de Frankel)" , ''Sém. Bourbaki 1979/80'' , ''Lect. notes in math.'' , '''842''' , Springer  (1981)  pp. 11–19</TD></TR>
</table>

====Comments====
The theorem stated in the last line of the text is due to S. Mori [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Mori,   "Positive manifolds with ample tangent bundles"  ''Ann. of Math.'' , '''110'''  (1979)  pp. 593–606</TD></TR></table>