  We prove that if a local ring admits a (pd-)test module of finite complete
intersection dimension, then it is a complete intersection ring. This answers,
positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, we
first investigate another question raised by Celikbas and Sather- Wagstaff
concerning ascent properties of (pd-)test complexes under weakly regular
homomorphisms.

Commutative Algebra