proper generator in a Grothendieck category

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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