%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: center of Abelian category
%%% Primary Category Code: 00.
%%% Filename: CenterOfAbelianCategory.tex
%%% Version: 12
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
Let $\mathcal{A}$ be an \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html}. Then one also has the identity morphism (or identity functor) $id_{\mathcal{A}} : \mathcal{A} \to \mathcal{A}$. One defines the {\em center of the Abelian category $\mathcal{A}$} by
$$Z(\mathcal{A}) = End(id_{\mathcal{A}}).$$
\end{definition}
\begin{example}
One can show that the center is $Z(CohX) \cong \mathcal{O}((X)$ for any \htmladdnormallink{algebraic variety}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} where $\mathcal{O}(X)$ is the ring of global \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} on $X$ and ${\bf Coh}(X)$ is the Abelian category of coherent sheaves over $X$.
\end{example}
One can show also prove the following lemma.
\begin{theorem} {\bf Associative Algebra Lemma}
If $A$ is a associative algebra then its center $$Z(A-mod) = ZA.$$
\end{theorem}
\end{document}
