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Gabriel-Popescu Theorem

The Gabriel-Popescu theorem states that every Grothendieck category is a localization of a module category. More explicitly, let A be a Grothendieck category with a generator U. This means that Hom(U,-) is a faithful functor. Let R=End(U) be its endomorphism ring. Then the functor

Hom(U,-):A->Mod-R,

is a fully faithful functor and has a left adjoint that is an exact functor. Fully faithful means that the functor gives a bijection between every Hom-set in A and the corresponding Hom-set in Mod-R. Consequently, A is equivalent to the quotient of Mod-R by a localizing subcategory, one closed under subobjects, quotient objects, extensions, and coproducts.

See also

Abelian Category, Exact Functor, Faithful Functor, Fully Faithful Functor, Grothendieck Category, Hom Functor, Module Category

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References

Popescu, N. Abelian Categories with Applications to Rings and Modules. London, England: Academic Press, 1973.The Stacks Project Authors. "The Gabriel-Popescu Theorem." https://stacks.math.columbia.edu/tag/0F5R.

Cite this as:

Weisstein, Eric W. "Gabriel-Popescu Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Gabriel-PopescuTheorem.html

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