MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up some background on homotopy theory.
Does the analogue of Hartshorne exist? Are there any must-reads for stable homotopy theory and spectra? What would your advice for a beginning graduate student be?
Just to set a starting point, I would ask for suggestions for someone who's familiar with a good chunk of most concepts covered in Hatcher and some differential geometry, but not much more.


I guess in the back of my mind is trying to understand some of this "brave new algebraic geometry", ie geometry over an $E\_\infty$-ring spectrum.



Why not a Roadmap for Homotopy Theory and Spectra?
There have been several questions previously in this vein. [This one](https://mathoverflow.net/questions/136077/an-advanced-beginners-book-on-algebraic-topology) asks for an advanced beginners book. The consensus seemed to be that it was difficult to find a one-size-fits-all text because people come in with such diverse backgrounds. Peter May's textbook [A Concise Course in Algebraic Topology](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf) is probably the closest thing we've got. If you like that, then you can also read [More concise algebraic topology](http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf) by May and Ponto. I also recommend [Davis and Kirk](http://indiana.edu/~lniat/m621notessecondedition.pdf)'s Lecture Notes in Algebraic Topology. I think these would be a very reasonable place for a beginning grad student to start (assuming they'd already studied Allen Hatcher's book or something equivalent).


Another question asked for [textbooks bridging the gap](https://mathoverflow.net/questions/18041/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap) and got similar answers. Finally, there was a more specific question about a [modern source for spectra](https://mathoverflow.net/questions/81740/modern-source-for-spectra-including-ring-spectra?rq=1) and this has a host of useful answers. Again, Peter May and coauthors have written quite a bit on the subject, notably EKMM for S-modules, Mandell-May for Orthogonal Spectra, and MMSS for diagram spectra in general. Another great reference is Hovey-Shipley-Smith Symmetric Spectra. On the more modern side, there's Stefan Schwede's Symmetric Spectra Book Project. All these references contain phrasing in terms of model categories, which seem indispensible to modern homotopy theory. Good references are Hovey's book and Hirschhorn's book.


Since you mention that you're especially interested in $E\_\infty$ ring spectra, let me also point out Peter May's survey article [What precisely are $E\_\infty$ ring spaces and $E\_\infty$ ring spectra?](http://arxiv.org/abs/0903.2813)


