Over a field, the stable module category of a finite group is obtained from all kG-modules by killing the projectives, which are also the injectives. Over a general commutative base ring k that identification fails, so the construction is rebuilt from two relative notions: a module is weakly projective (equivalently weakly injective, by Higman's criterion) when it is a summand of one induced from the trivial subgroup, and the weakly injectives are the injective objects of a Frobenius exact structure whose stable category is triangulated. Enlarging this class to the cw-injectives - the filtered colimits of weakly injectives - yields a compactly generated tensor triangulated category whose compact objects are the finitely presented modules. Also covers the recollement relating this category to the pure derived category of kG, which over a non-field base is genuinely coarser than the ordinary derived category; base change along k to k' and the resulting local-global principle over the prime spectrum; finite generation of relative cohomology over a noetherian base; and the way the Balmer spectrum of the resulting tensor triangulated category can be disconnected.
For a finite group G and a field k, the stable module category StMod(kG) is built from all kG-modules by identifying two homomorphisms whenever their difference factors through a projective module; i…