PlanetPhysics/Proper Generator in a Grothendieck Category

Introduction: family of generators and generator of a category
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 $$ \alpha u \neq 0$$.

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
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 $$\iota$$, $$Hom_{\mathcal{G}}(U_p,U_p) \cong Hom_{\mathcal{G}} (U',U_p),$$ if and only if $$i$$ is an isomorphism.

\begin{theorem} Any commutative ring is the endomorphism ring of a proper generator in a suitably chosen Grothendieck category. \end{theorem}