# PlanetPhysics/Center of Abelian Category

Let ${\displaystyle {\mathcal {A}}}$ be an abelian category. Then one also has the identity morphism (or identity functor) ${\displaystyle id_{\mathcal {A}}:{\mathcal {A}}\to {\mathcal {A}}}$. One defines the center of the Abelian category $\displaystyle \mathcal{A$ } by ${\displaystyle Z({\mathcal {A}})=End(id_{\mathcal {A}}).}$

One can show that the center is ${\displaystyle Z(CohX)\cong {\mathcal {O}}((X)}$ for any algebraic variety where ${\displaystyle {\mathcal {O}}(X)}$ is the ring of global regular functions on ${\displaystyle X}$ and ${\displaystyle {\mathbf {C} oh}(X)}$ is the Abelian category of coherent sheaves over ${\displaystyle X}$.

One can show also prove the following lemma. \begin{theorem} {\mathbf Associative Algebra Lemma}

If ${\displaystyle A}$ is a associative algebra then its center ${\displaystyle Z(A-mod)=ZA.}$ \end{theorem}