= Failure of ordinary sheaf-dual Serre duality for a skyscraper sheaf
On $\mathbb P_k^n$ with $n>0$, let $\mathcal F=i_*k$ at a rational point. It has one-dimensional <global sections>, but its <dual of a sheaf> is zero: a local map from the residue field into the local polynomial ring must have image annihilated by a nonzero parameter, which is impossible in an <integral domain>. Thus $H^n(\mathcal F^\vee(-n-1))=0$. <Serre duality> for arbitrary coherent sheaves instead uses an <Ext functor>; replacing this by the ordinary dual can lose torsion information.
Back to article page
