Splitting Monoidal Stable Model Categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S, S] . An idempotent e of this ring will split the homotopy category: [X,Y ] ∼= e[X,Y ]⊕(1−e)[X,Y ] . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to LeSC × L(1−e)SC and [X,Y ] LeSC ∼= e[X,Y ] . This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.