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Codisjunctors and singular epimorphisms in the category of commutative rings. (English) Zbl 0665.18008

In this paper, the author presents a new universal construction of objects of fractions, a.o. in the category of commutative algebras. The starting point is the notion of codisjunctors. A pair of morphisms (g,h): \(C\rightrightarrows A\) is codisjoint if any morphism u: \(A\to X\) with \(ug=uh\) has as its target a final object. If (fg,fh) is codisjoint for some pair (g,h): \(C\rightrightarrows A\) and some morphism f: \(A\to B\), then one says that f codisjoints (g,h). Finally, a codisjunctor of (g,h): \(C\rightrightarrows A\) will be a universal morphism which codisjoints (g,h); if such a codisjunctor exists, we say that (g,h) is codisjunctable.
In the category of commutative rings with unit, one extends the terminology to ideals of a ring R by identifying an ideal I with the pair of canonical projections \((r_ 1,r_ 2): R\rightrightarrows A\), where R is the congruence relation modulo I on A. In this category, it is thus clear that one may restrict to the study of codisjunctable ideals and their codisjunctors.
One may then prove that any projective ideal of finite type I of A is codisjunctable and has as a codisjunctor the canonical localization morphism \(A\to \lim Hom_ A(I^ n,A)\), where the second member is just the localization of A at the I-adic Gabriel topology. The author also shows how this result may be extended to so-called n-projective ideals. In fact, he proves that an ideal of finite type with zero annihilator is codisjunctable if and only if it is n-projective for some integer n. Moreover, it appears that an A-ideal I is codisjunctable exactly when the open set D(I)\(\subset Spec(A)\) is affine. If one calls singular epimorpism a morphism which is the codisjunctor of an ideal, then one may show that these are just the epimorphisms f: \(A\to B\) which make B into a finitely presented flat A-algebra, i.e. the “rings of quotients” of A. In the final part of the paper, the author shows how these notions may be generalized to sets of pairs of morphisms (g,h): \(C\rightrightarrows A\), thus defining the notion of simultaneous codisjunctor. Using this, he essentially proves that all perfect localizations of a ring (in the sense of Gabriel) may be recovered as a simultaneous codisjunction.
Reviewer: A.Verschoren

MSC:

18E40 Torsion theories, radicals
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References:

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