Flat ring epimorphisms and universal localizations of commutative rings

Abstract

We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

Derived decompositions of abelian categories I

“…This implies that λ ⊗ R E(R p) is always surjective, and therefore λ ⊗ R I is surjective. Thus Corollary 4.3 follows from Proposition 4.2 (2). ◻ Next, we show that the derived decompositions in Propositions 4.1 and 4.2 provide also lower half recollements of derived categories.…”

“…Note that S S ∈ P(X ), the category of projective S-modules. If (X ,Y) is a complete Ext-orthogonal pair in A, then projdim( R S) ≤ 1 by Corollary 3.8 (2). This shows the necessity of (1).…”

“…The proof is divided into two parts. The first one is for the proof of Theorem 1.2(1), while the second one is for that of Theorem 1.2 (2). In Section 4 we apply Theorem 1.2 to construct derived decompositions of module categories from various aspects: ring epimorphisms, localizing subcategories, commutative noetherian rings and nonsingular rings.…”

“…Then S is commutative and flat as an Rmodule, and therefore λ is a homological ring epimorphism. By Proposition 4.2 (2), it suffices to show Coker(λ) ⊗ R I = 0 (or equivalently, λ ⊗ R I is surjective) for any injective R-module I.…”

“…Theorem 1.2 (2) implies that if A has enough projectives and injectives, then the existences of D *decompositions of A for all * ∈ {b,+,−,∅} are equivalent. This applies to the module categories of rings.…”