PlanetPhysics/Center of Abelian Category

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

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

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

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