One knows that the support $S$ of a coherent sheaf on a noetherian scheme is closed.
E.g. on an affine scheme $X=Spec(A)$ and $F$ corresponding to a finitely generated $A$-module $M$, then the closed subset which corresponds to $S$ is just $V(Ann(M))$.
One often says that $S$ is endowed with the structure of a closed subscheme by taking the sheaf of ideals $Ann(F)$ defined as the kernel of $\mathcal O_X \rightarrow Hom_{\mathcal O_X}(F,F)$.
Now my question: this is not the (unique) reduced subscheme structure, isn't it?
Can one anyhow describe the reduced structure?
Thanks
Best Answer
No it isn't the reduced induced closed subscheme structure in general. For example, let $A={\bf Z}$, $M={\bf Z}/4{\bf Z}$. Then ${\rm Ann}(M)=4{\bf Z}$ and the prime ideal defining $S$ (with its reduced structure) is $2{\bf Z}={\rm rad}({\rm Ann}(M))$. So if $S$ is endowed with the reduced structure, it is isomorphic to ${\rm Spec}({\bf Z}/2{\bf Z})$ and if it is endowed with the structure given by the annihilator then it is isomorphic to ${\rm Spec}({\bf Z}/4{\bf Z})$.