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We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.

I was wondering if there is a SS for arbitrary triangulated categories of which this is special case. Ravenel talks about generalizing this to other cohomology theories while still staying in the category of spectra.

More specifically I am curious if we assume our category to have enough projectives or injectives can we avoid invoking the smash product?

I am new to Stable Homotopy theory and use smash products as black box. It would be really delightful if I could replace this by projectives or fibered products or something similar but algebraic. Of course constructing a projective cover must involve smash products but that is for another day.

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