center of Abelian category

Definition 0.1   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

$\displaystyle Z(\mathcal{A}) = End(id_{\mathcal{A}}).$

Example 0.1   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 $ {\bf Coh}(X)$ is the Abelian category of coherent sheaves over $ X$ .

One can show also prove the following lemma.

Theorem 0.1   Associative Algebra Lemma

If $ A$ is a associative algebra then its center

$\displaystyle Z(A-mod) = ZA.$



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