Homological ring epimorphisms and recollements from exact pairs. I

Details Homological ring epimorphisms and recollements from exact pairs. I Homological ring epimorphisms and recollements from exact pairs. I

2012-03-23

arxiv.org/abs/1203.5168v1

Mathematics

Representation Theory

37 pages

Scientific paper

Homological ring epimorphisms are often used in modern representation theory and algebraic $K$-theory. In this paper, we give some new characterizations of when a universal localization related to an `exact' pair of ring homomorphisms is homological. These characterizations are flexible and applicable to many cases, thus give rise to a wide variety of new recollements (of derived module categories) which have become of interest in and attracted increasing attentions towards to understanding invariants in algebra and geometry. As a consequence, we show that if $\lambda: R\ra S$ is an injective homological ring epimorphism between commutative rings $R$ and $S$, then the derived module category of the endomorphism ring of the $R$-module $S\oplus S / R$ always admits a recollement of the derived module categories of $R$ and the tensor product $S\otimes_R End_R(S/R)$. In particular, this result is applicable to localizations of integral domains by multiplicative sets in commutative rings.

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