Tilting Modules Under Special Base Changes

Given a non-unit, non-zero-divisor, central element \(x\) of a ring \(\Lambda\), it is well known that many properties or invariants of \(\Lambda\) determine, and are determined by, those of \(\Lambda / x \Lambda\) and \(\Lambda_x\). In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over \(\Lambda\) gives rise to tilting modules over \(\Lambda_x\) and \(\Lambda / x \Lambda\) after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over \(\Lambda\) is tilting if its corresponding localization and quotient are tilting over \(\Lambda_x\) and \(\Lambda / x \Lambda\) respectively.