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center of Abelian category (Definition)
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

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



"center of Abelian category" is owned by bci1.
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See Also: Abelian category

Also defines:  identity functor, identity morphism, associative algebra lemma
Keywords:  center, Abelian category, ring of global regular functions on $X$, algebraic variety

Cross-references: functions, regular, algebraic, abelian category
There are 3 references to this object.

This is version 12 of center of Abelian category, born on 2009-06-15, modified 2009-06-15.
Object id is 805, canonical name is CenterOfAbelianCategory.
Accessed 1197 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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