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Relative normalizing extensions. (English) Zbl 0856.16023

Under certain conditions a homomorphism \(f:R\to S\) of commutative noetherian rings induces a morphism \(\text{Spec}(S)\to\text{Spec}(R)\) between the associated affine schemes (essentially by taking inverse images), this is no longer true, even just set-theoretically, over noncommutative rings. The main reason for this is that the inverse image of a prime ideal is in general no longer a prime ideal. In order to remedy this, one has to impose on \(f\) some extra conditions, such as being a centralizing extension, for example.
In the present paper, the author restricts to ring morphisms \(f\), which do not necessarily induce a global map between the corresponding affine schemes, but do when restricted to suitable so-called generically closed subsets, the latter occurring in a natural way when studying local behaviour of rings and modules outside of the strict noetherian context.
The relative normalizing extensions introduced in this paper behave nicely with respect to localization and yield a wide class of morphisms, which induce functorial behaviour between the associated generically closed subsets endowed with a suitable structure sheaf, thus providing the basic machinery to further develop relative noncommutative algebraic geometry.

MSC:

16S20 Centralizing and normalizing extensions
16S60 Associative rings of functions, subdirect products, sheaves of rings
14A22 Noncommutative algebraic geometry
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18E35 Localization of categories, calculus of fractions
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