Local cohomology and Foxby classes

Let R be a commutative Noetherian ring and I a proper ideal of R. In this paper, we study finitely generated R-modules M with only one non-vanishing local cohomology module \({H}_I^c(M)\) $H_I^c(M)$ where \(c=cd(I,M)\) $c=cd(I,M)$ . Let C be a semidualizing R-module. We investigate the conditions under which \({H}_I^c(M)\) $H_I^c(M)$ belongs to either the Auslander class \(\mathscr{A}_C(R)\) ${\mathcal{A}}_C(R)$ or the Bass class \(\mathscr{B}_C(R)\) ${\mathcal{B}}_C(R)$ .