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The main result proved here is that      B P ∗  ( E G ×  G X ) {\text {B}}{{\text {P}}_* }(EG{ \times _G}X)       has finite homological dimension when      G = Z p G = {{\mathbf {Z}}_p}       and      X X       is a finite      G  - CW G{\text { - CW}}      -complex. The argument uses      B P ∗  BP {\text {B}}{{\text {P}}_*}{\text {BP}}      -comodules.

Finite homological dimension of ${\rm BP}\sb\ast (X)$ for infinite complexes