Finiteness of complete intersection dimensions of RHom complexes and Ext modules

In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules $M$ over a Noetherian local ring $R$ such that ${\mathrm{Hom}}_R(M,R)$ or ${\mathrm{Hom}}_R(M,M)$ has finite complete intersection injective dimension, and provide Gorenstein criteria.