Tilting Modules Under Special Base Changes

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