6
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thanks a lot for all the answers, this is exactly what I was looking for. $\endgroup$
    – apurv
    May 26, 2014 at 14:21

3 Answers 3

7
$\begingroup$

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

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.