proper generator in a Grothendieck category

0.1 Introduction: family of generators and generator of a category

Definition 0.1. Let 𝒞 be a category. A family of its objects {U }
i _i∈I is said to be a family of generators of 𝒞 if for every pair of distinct morphisms α,β : A → B there is a morphism u : U_i → A for some index i ∈ I such that αu≠βu.

One notes that in an additive category, {Ui} _i∈I is a family of generators if and only if for each nonzero morphism α in 𝒞 there is a morphism u : U_i → A such that αu≠0.

Definition 0.2. An object U in 𝒞 is called a generator for 𝒞 if U ∈ {Ui} _i∈I with {Ui } _i∈I being a family of generators for 𝒞.

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

0.2 Proper generator of a Grothendieck category

Definition 0.3. A proper generator U_p of a Grothendieck category 𝒢 is defined as a generator U_p which has the property that a monomorphism i : U′ → U_p induces an isomorphism ι,

Hom (U ,U ) ∼ Hom (U ′,U ), 𝒢 p p = 𝒢 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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