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}