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Functor \((\overline{S})^{-1}( )\) and Functorial Isomorphisms

  • Daouda FayeEmail author
  • Mohamed Ben MaaouiaEmail author
  • Mamadou SanghareEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1133)

Abstract

The main result of this paper is the following:

Let B be a unitary noetherian ring, \(A = Z(B)\), the center of B, S a central saturated multiplicative subset of A satisfying the left (respectively right) conditions of Ore (respectively the subset of regular elements of \(A-P\) where P is a prime ideal of A), \(S^{-1}A\) the ring of fractions of A in S, \((\overline{S})^{-1}B\) the ring of fractions of B in \(\overline{S}\), \({}_{B}M{}_{A}\) a free \((B-A)-\) bimodule of finite type, \(A-Mod_{ff}\) (respectively \(S^{-1}A-Mod_{ff}\)) the subcategory of \(A-Mod\) (respectively \(S^{-1}A-Mod\)) containing the free A-modules of finite type (respectively the free \(S^{-1}A\)-modules of finite type); then the covariant functors:

\(Ext_{(\overline{S})^{-1}B}^{n} (S^{-1} M,-):(\overline{S})^{-1}B-Mod_{ff} \rightarrow S^{-1}A-Mod_{ff}\) and \(Tor_{n}^{{S}^{-1}}A ({{S}^{-1}} M,-):{S}^{-1}A-Mod_{ff} \rightarrow (\overline{S})^{-1}B-Mod_{ff} \) are adjoint.

Keywords

Ring Left (right) conditions of Ore Closed multiplicative subset Letf A-module Ring of fractions Module of fractions Categories A-Mod Mod-A \(A-Mod_{ff}\) \(S^{-1}A-Mod_{ff}\) Functors \(S^{-1} ( )\) Ext and Tor 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Laboratory of Algebra, Cryptography and Algebraic Geometry and Applications – LACGAAUniversity Cheikh Anta Diop of DakarDakarSenegal
  2. 2.Laboratory of Algebra, Codes and Cryptography Applications (ACCA), UFR SATUniversity Gaston Berger (UGB) - St. Louis SenegalSaint LouisSenegal
  3. 3.Doctoral School of Mathematics-Computer – UCAD-SénégalUniversity Cheikh Anta Diop of DakarDakarSenegal

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