Depth formula for modules of finite reducing projective dimension

Abstract

We prove that the depth formula holds for two finitely generated Tor-independent modules over Cohen–Macaulay local rings if one of the modules considered has finite reducing projective dimension (for example, if it has finite projective dimension, or the ring is a complete intersection). This generalizes a result of Bergh–Jorgensen which shows that the depth formula holds for two finitely generated Tor-independent modules over Cohen–Macaulay local rings if one of the modules considered has reducible complexity and certain additional conditions hold. Each module that has reducible complexity also has finite complexity and finite reducing projective dimension, but not necessarily vice versa. So a new advantage we have is that, unlike modules of reducible complexity, Betti numbers of modules of finite reducing projective dimension can grow exponentially.