The Functor $$S^{-1}_C()$$ and Its Relationship with Homological Functors $$T or_n$$ and $$\overline{EXT}^n $$

Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A - Mod)

“…12, No. 4;2020 Then S −1 C () is an exact covariant functor. Proof see (Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)), proposition 1 Definition and proposition 2.3:…”

“…Now since on the one hand Hom • (X, −) and EXT 0 Comp(A−Mod) (X, −), where EXT n is considered to be the n-th funtor derived of Hom • , are isomorphic and on the other hand X − and T or Comp(B−Mod) 0 (X, −), where T or Comp(B−Mod) n is the n-th derived functor of the tensor product functor X −, are isomorphic then we can conclude that EXT 0 Comp(A−Mod) (X, −) and T or Comp(B−Mod) 0 (X, −) are adjoint functors. Besides, in (Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)) we showed that the functor S −1 C () commute with the functors tensor product, Hom • , EXT n and T or n on the objects. So, the question is of course this: if we can have the generalization of that results.…”

“…Then S −1 C Hom • (X, −) and Hom • (S −1 C (X), S −1 C ()) are isomorphic. Proof we know that according to the proof of theorem 7 in [Dembele, B., Maaouia, B.,F., & Sanghare, M. (2020)] that for all complex of left A modules Y there exist an isomorphism Φ X,Y :…”

“…12, No. 4;2020 Now let f : Y 1 −→ Y 2 be a map of complexes, let us show the commutativity of the following diagram:…”