Strongly $n$-AIR-tilting modules

We introduce the notion of (strongly) $n$-AIR-tilting modules, which is a high dimension version of support $\tau$-tilting modules. The relations between them and $n$-silting modules and $n$-quasi-tilting modules, as well as generalized two-term silting complexes, are investigated. Our results particularly suggest a way to negate the rank question for silting complexes.