# Adams Spectral Sequence for Triangulated Categories

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 compactly generated triangulated categories of which this is special case.

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 think of smash products as black box. It would be really delightful if I could replace having a smash product by having enough projectives or something similar but algebraic.

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Thanks a lot for all the answers, this is exactly what I was looking for. – apurv May 26 '14 at 14:21

In addition to the notes of Haynes Miller see http://jdc.math.uwo.ca/papers/ideals.pdf

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These course notes by Haynes Miller seem to be doing exactly what you ask. In the general case the smash product is replaced by a symmetric monoidal structure, which has to interact well with the triangulated structure.

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A general treatment of the Adams spectral sequence in the context of triangulated categories based on the work of Brinkmann and Christensen can be found here: http://arxiv.org/pdf/0801.1344.

No monoidal structure is necessary (although, in applications, you might use a monoidal structure to define a homological ideal).

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