The Gabriel Popescu theorem states that any Grothendieck category \ mathcal { A } is equivalent to a full subcategory of the category \ operatorname { Mod } ( R ) of right modules over some unital ring R ( which can be taken to be the endomorphism ring of a generator of \ mathcal { A } ), and \ mathcal { A } can be obtained as a Serre quotient of \ operatorname { Mod } ( R ) by some localizing subcategory.
42.
For a Frobenius algebra extension A | B ( such as A and B group algebras of a subgroup pair of finite index ) the two one-sided conditions of depth two are equivalent, and a notion of depth n > 2 makes sense via the right endomorphism ring extension iterated to generate a tower of rings ( a technical procedure beyond the scope of this survey, although the first step, the endomorphism ring theorem, is described in the section on Frobenius extension under Frobenius algebra ).
43.
For a Frobenius algebra extension A | B ( such as A and B group algebras of a subgroup pair of finite index ) the two one-sided conditions of depth two are equivalent, and a notion of depth n > 2 makes sense via the right endomorphism ring extension iterated to generate a tower of rings ( a technical procedure beyond the scope of this survey, although the first step, the endomorphism ring theorem, is described in the section on Frobenius extension under Frobenius algebra ).
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