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

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.

Submission 17 A2C Conference.