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proper generator in a Grothendieck category (Topic)

Introduction: family of generators and generator of a category

Definition 0.1   Let $\mathcal{C}$ be a category. A family of its objects $\left\{U_i\right\}_{i \in I}$ is said to be a family of generators of $\mathcal{C}$ if for every pair of distinct morphisms $\alpha, \beta: A \to B $ there is a morphism $u: U_i \to A$ for some index $i \in I$ such that $\alpha u \neq \beta u$.

One notes that in an additive category, $\left\{U_i\right\}_{i \in I}$ is a family of generators if and only if for each nonzero morphism $\alpha$ in $\mathcal{C}$ there is a morphism $u: U_i \to A$ such that $\alpha u \neq 0$.

Definition 0.2   An object $U$ in $\mathcal{C}$ is called a generator for $\mathcal{C}$ if $U \in \left\{U_i\right\}_{i \in I}$ with $\left\{U_i\right\}_{i \in I}$ being a family of generators for $\mathcal{C}$.

Equivalently, (viz. Mitchell) $U$ is a generator for $\mathcal{C}$ if and only if the set-valued functor $H^U$ is an imbedding functor.

Proper generator of a Grothendieck category

Definition 0.3   A proper generator $U_p$ of a Grothendieck category $\mathcal{G}$ is defined as a generator $U_p$ which has the property that a monomorphism $i: U' \to U_p$ induces an isomorphism $\iota$,

$\displaystyle Hom_{\mathcal{G}}(U_p,U_p) \cong Hom_{\mathcal{G}} (U',U_p),$
if and only if $i$ is an isomorphism.
Theorem 0.1   Any commutative ring is the endomorphism ring of a proper generator in a suitably chosen Grothendieck category.



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Cross-references: commutative ring, isomorphism, monomorphism, Grothendieck category, functor, additive category, morphisms, generators, objects, category

This is version 3 of proper generator in a Grothendieck category, born on 2009-02-02, modified 2009-02-02.
Object id is 470, canonical name is ProperGeneratorInAGrothendieckCategory.
Accessed 320 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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