  We consider the relationship between the relative stable category of Benson,
Iyengar, and Krause and the usual singularity category for group algebras with
coefficients in a commutative noetherian ring. When the coefficient ring is
self-injective we show that these categories share a common, relatively large,
Verdier quotient. At the other extreme, when the coefficient ring has finite
global dimension, there is a semi-orthogonal decomposition, due to Poulton,
relating the two categories. We prove that this decomposition is partially
compatible with the monoidal structure and study the morphism it induces on
spectra.
