When geometrically flavoured words like "mapping cone" or "chain homotopy" crop up in homological algebra, there's usually a good reason. (In this case, looking at the chain complexes associated with each geometric construction gives the algebraic construction).

But, for the life of me, I cannot find a reason why the "stable module category" deserves to be called "stable." It doesn't seem, superficially at least, to be related to the stable homotopy category. So:

Are these categories related in any nontrivial manner that gives merit to the terminology?

If not, is there some other reason why the stable module category deserves to be called stable? (Mind you, this sounds like it might have a higher-categorical answer, and my knowledge of higher category theory is very sketchy... so do be gentle if that's the direction your answer is taking!)

EDIT: The stable module category for a ring $R$ has $R$-modules for objects and maps "modulo projectives" for morphisms. That is, we put an equivalence relation on $Hom_{R}(M,N)$ by declaring $f \sim 0$ if $f$ factors through a projective module. For certain rings (e.g. group rings), this category is tensor triangulated (which is why I'm interested).

stably, that is, up to adding something to both sides) $\endgroup$5more comments