Virtual projectivity and the stable module category

Abstract

We introduce the concept of a nearly null map in the stable module category, and relate it to the notion of virtual projectivity. We show that the trivial module k is virtually M-projective if and only if the map f:N→k in the triangle N→fk→M⊗M⁎ is nearly null. If these conditions hold, then M generates the stable module category in fewer than ℓ steps, where ℓ is the radical length of the group algebra. We give examples to show that if M generates the stable module category then k is not necessarily virtually M-projective. We also give examples to show that for a given group, the degree of virtual projectivity of k with respect to a module M is not bounded. Finally, we develop a theory of virtual M-variety of a module X. This is a subset of the spectrum of the cohomology ring which controls virtual projectivity of X with respect to M.