Tilting Modules Under Special Base Changes

TL;DR

This paper studies how the property of being a tilting module is preserved and characterized under certain base changes involving localization and quotient by a central element in a ring.

Contribution

It establishes that tilting modules over a ring induce tilting modules over localized and quotient rings, and under mild conditions, the converse holds.

Findings

Tilting modules over mbda induce tilting modules over mbda_x and mbda/xmbda.

Under mild conditions, tilting modules over mbda are characterized by their localizations and quotients.

The paper provides conditions under which tilting properties are preserved under base change.

Abstract

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.