EXTENSION AND TORSION FUNCTORS WITH RESPECT TO SERRE CLASSES

In this paper we generalize the Rigidity Theorem and Zero Divisor Conjecture for an arbitrary Serre subcategory of modules. For this purpose, for any regularM-sequence x۱; :::; xn with respect to S we prove that if TorR ۲ ((x۱;:::;x R n); M) ۲ S, thenTorR i ((x۱;:::;x R n); M) ۲ S, for all i ≥ ۱. Also we show that if Extn R+۲((x۱;:::;x R n); M) ۲ S,then Exti R((x۱;:::;x R n); M) ۲ S, for all integers i ≥ ۰ (i ̸= n).

Keywords:

Serre classes , Zero Divisor Conjecture , RigidityTheorem , Top local cohomology