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Localization Functor S-1() and Functors ext and tor

Daouda Faye and Mohamed Ben Faraj BEN MAAOUIA
Associated member, Directory of Algebraic Laboratory,
Cryptography, Codes and Applications (LACCA)- UFR SAT
Université Gaston Berger, Saint-Louis (UGB), Sénégal

Member, Directory of Algebraic Laboratory,
Cryptography, Algebraic Geometry and Applications
LACGAA - Technologies and Sciences Faculty
Université Cheikh Anta Diop, Dakar (UCAD), Sénégal

Mathematics Subject Classification: (2010)
Key words: Ring, duo-ring, left (right) Ore’s conditions, multiplicatively closed subset, left A-module, ring of fractions, module of fractions, category A-Mod, Mod-A, functors S-1(), Ext and Tor.


In this paper, unless otherwise stated, B is a duo-ring, A a subring of B, S a multiplicatively closed subset of the ring A satisfying the left (right respectively) Ore’s conditions, S the set of regular elements of S, AM a left A-module, BNA a (B -A)-bimodule (respectively ANB a (A-B)-bimodule), S-1A the ring of fractions of A on S ; particularly, if A is a duo-ring and P is a prime ideal of A then S is the set of regular elements of A - P.
We establish the following results :

1) Isomorphisms of left (S)-1B-modules :

(S-)- 1Hom  (M, N) ~= Hom  -1 (S-1M, S-1N ))
         A             S  A

where M is an (A - B)-bimodule and N a left A-module ;

2) Isomorphisms of left (S)-1B-modules :

 --                   -1
(S)-1TorAn(M, N ) ~= TorSn A(S-1M, S-1N )  and

   A        ~    AP
Torn(M, N)P = Torn  (MP ,NP );

3) Isomorphisms of right (S)-1B-modules :

 --                       (           )
(S)-1ExtnA (M, N) ~= ExtnS-1A S- 1M, S -1N    and

   n        ~    n
ExtA(M, N)P = ExtAP (MP  ,NP );

4) If B is noetherian, BMA a (B - A)-bimodule, then the functors :

Extn-- 1 (S -1M,- ) : (S)-1B - M od → S-1A - M od and
   (S) B

TornS-1A (S-1M,-) : S-1A - Mod (S)-1B - Mod are adjoint.

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