zero divisor conjecture

It is proved that if $sum_{gin G} a_g g$ is a non-zero zero divisor element of the complex group  algebra $mathbb{C}G$ of a torsion-free group G then  $2sum_{gin G} |a_g|^2<big( sum_{gin G} |a_g|big)^2$.

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 x1; :::; xn with respect to S we prove that if TorR 2 ((x1;:::;x R n); M) 2 S, thenTorR i ((x1;:::;x R n); M) 2 S, for all i ≥ 1. Also we show that if Extn R+2((x1;:::;x R n); M) 2 S,then Exti R((x1;:::;x R n); M) 2 S, for all integers i ≥ 0 (i ̸= n).