On Bhargava rings

Chems-Eddin, Mohamed Mahmoud; Ouzzaouit, Omar; Tamoussit, Ali

doi:10.21136/mb.2022.0137-21

Math.Boh. 2022

DOI: 10.21136/mb.2022.0137-21

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On Bhargava rings

Mohamed Mahmoud Chems-Eddin¹,

Omar Ouzzaouit²,

Ali Tamoussit³

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2024

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The Local Picard Group of a Ring Extension

Spirito

2024

Mediterr. J. Math.

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Given an integral domain D and a D-algebra R, we introduce the local Picard group $$\textrm{LPic}(R,D)$$ LPic ( R , D ) as the quotient between the Picard group $$\textrm{Pic}(R)$$ Pic ( R ) and the canonical image of $$\textrm{Pic}(D)$$ Pic ( D ) in $$\textrm{Pic}(R)$$ Pic ( R ) , and its subgroup $$\textrm{LPic}_u(R,D)$$ LPic u ( R , D ) generated by the the integral ideals of R that are unitary with respect to D. We show that, when $$D\subseteq R$$ D ⊆ R is a ring extension that satisfies certain properties (for example, when R is the ring of polynomial D[X] or the ring of integer-valued polynomials $$\textrm{Int}(D)$$ Int ( D ) ), it is possible to decompose $$\textrm{LPic}(R,D)$$ LPic ( R , D ) as the direct sum $$\bigoplus \textrm{LPic}(RT,T)$$ ⨁ LPic ( R T , T ) , where T ranges in a Jaffard family of D. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of D.

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The Local Picard Group of a Ring Extension

Spirito

2024

Mediterr. J. Math.

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show abstract

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On Bhargava rings

Cited by 1 publication

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The Local Picard Group of a Ring Extension

The Local Picard Group of a Ring Extension

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