Weak Silting Modules

It is well-established that weak n-tilting modules serve as generalizations of both n-tilting and n-cotilting modules. The primary objective of this paper is to delineate the characterizations of weak n-silting modules and elaborate on their applications. Specifically, we aim to establish the "triangular relation" within the framework of silting theory in a module category, and provide novel characterizations of weak n-tilting modules. Furthermore, we delve into the properties of n-(co)silting modules and their interrelations with some other types of modules. Additionally, we explore the conditions under which a weak n-silting module can be classified as partial n-silting, weak n-tilting, or partial n-tilting. Notably, we establish and prove that Bazzoni’s renowned characterization of pure-injectivity for cotilting modules remains valid for weak n-silting modules with respect to \(\mathcal {F}_{\mathbb {T}}\) ${\mathcal{F}}_{\mathbb{T}}$ . Lastly, we investigate weak n-silting and weak n-tilting objects in a morphism category.