Gorenstein weak \(n\)-silting modules and weak \(n\)-star modules (Q6071315)

Let \(\mathbf{W}\) be a left \(n\)-Gorenstein flat resolution of a left \(R\)-module \(W\). The authors define the class \(\mathcal F_{\mathbf W}\) as those right \(R\)-modules \(X\) such that \(X\otimes {\mathbf W}\) is exact. In Section 3, the authors study the properties of this class \(\mathcal F\). Next section is devoted to the study of Gorenstein weak \(n\)-silting modules. A left \(R\)-module is called a Gorenstein weak \(n\)-silting module with respect to \(\mathbf W\), if \({\mathcal F}_{\mathbf W}=\mathrm{Cogen}_G^n(W^+)\), where \(\mathrm{Cogen}_G^n(M)\) is the class of left \(R\)-modules \(X\) that are \(n\)-copresented by \(M\), i.e. admitting an exact sequence \(0\to X\to M^1\to \cdots M^n\) with each \(M^i\) a direct summand of a direct product of \(M\), and \(W^+=\mathrm{Hom}_{\mathbf Z}(M,\mathbb{Q}/\mathbb{Z})\) is the character module of \(M\). Several properties of this class of modules are obtained. A similar result of the triangular relation in the setting of tilting theory [\textit{X. Zhang} and \textit{L. Yao}, Commun. Algebra 37, No. 12, 4316--4324 (2009; Zbl 1187.16009)] is given. In the last section, the authors introduce the notion of Gorenstein weak \(n\)-star modules. It is shown that all Gorenstein weak \(n\)-silting modules are Gorenstein weak \(n\)-star modules and the converse is true if the class of Gorenstein flat modules is contained in \(\mathrm{Cogen}_G^n(T^+)\).