This is a list useful reading on homotopy theory in algebraic geometry. For a lot more resources, see the resource page maintained by Aravind Asok
Summary of background in algebraic geometry: notes by Marc Levine.
Many useful notes, including an introduction to sheaf cohomology, can be found at The Rising Sea
Notes by Fesenko.
Notes by Henning Krause.
A very readable introduction by Dwyer and Spalinski.
A guide by Cheng and Lauda.
The excellent book by May should cover everything.
A nice background paper by Dundas
Some slides by Strickland on stable homotopy
See reading list on page 2 of Strickland’s bestiary
DEA thesis by Joel Riou.
The original paper by Morel and Voevodsky
Notes from Tyler Lawson’s page
Voevodsky’s ICM lecture
Gillet in K-theory handbook, section 2.5.
Many nice things are in these notes by Dundas. More generally, check the Nordfjordeid summer school volume on motivic homotopy theory. Available in folder AG/Motives.
Talk in Toronto by Levine
See lectures of Levine at the Asian-French summer school
Voevodsky: Open problems I
Slides of Jardine
Levine: The homotopy coniveau filtration. (Looks very nice) See also Chow’s moving lemma and the homotopy coniveau tower.
Check all other papers of Voevodsky!
Weibel’s road map
Hornbostel on motivic chromatic homotopy theory.
Po Hu on the Picard group, and on S-modules in the stable homotopy category of schemes. Here is also something on the Steinberg relation
Check the paper on Motivic functors by Bjørn Ian Dundas, Oliver Röndigs, Paul Arne Østvær.
Motivic cell structures , by Daniel Dugger and Daniel C. Isaksen
Levine slides on Postnikov towers. See by the way everything on Levine’s web page, for example this
Biedermann: L-stable functors. “This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product.”
Weight structures, weight filtrations, weight spectral sequences, and weight complexes (for motives and spectra) , by Mikhail V. Bondarko: http://www.math.uiuc.edu/K-theory/0843
Joseph Ayoub thesis, on the six operations formalism in the stable homotopy category. Also: Ayoub in Nagel and Peters: nice exposition of various categories.
Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas, by Joël Riou
Half-page intro by Grayson, K-th handbook p 63.
Short intro by Kahn in K-theory handbook, pp374 (a possible starting point for rewriting stuff).
Joel Riou on the stable homotopy category of a site with interval. See also http://www.math.uiuc.edu/K-theory/0825
Levine Toen Jardine Morel Voevedsky Kahn
Can one work with infinite loop space machines in algebraic geometry? Can this lead to a recognition principle?
Some Open questions - a summary from a conference in Palo Alto
nLab page on Motivic homotopy theory reading