I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic topology, homotopy theory, and algebraic $K$-theory. I would like to stress that this is a broad interest, and I am not exactly sure what to look at. One program in this intersection that has recently caught my eye is the homotopy theory of schemes and in particular motivic homotopy theory. Another program that has interested me is derived algebraic geometry.
I have looked at the following posts for guidance but still feel a little overwhelmed with the breadth of these areas of research.
About my background:
Of the relevant topics, I feel the most comfortable with algebraic geometry. I have some familiarity with classical varieties, schemes, and sheaf cohomology (via Hartshorne and large portions of Vakil’s notes). I am secondly most comfortable with algebraic topology (at the level of Hatcher’s book) and homotopy theory (May’s Concise Course in Algebraic Topology). I am also roughly familiar with some topological $K$-theory at the level of Milnor-Stasheff.
On the other hand, my understanding of stable homotopy theory and higher algebra/homotopical algebra is quite weak. I am also not familiar with the modern weaponry in algebraic geometry such as stacks and étale cohomology.
Question(s). I would like some suggestions as for:
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What directions should I move toward given my interests?
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Roadmaps and references for those directions, and in particular derived algebraic geometry and motivic homotopy theory given my background.
Best Answer
This sounds like a very exciting phase of your studies! As you point out, there's plenty of intersections between algebraic geometry and homotopy theory. One intersection is motivic homotopy theory, and I think of derived algebraic geometry (DAG) as a different point of intersection, which can also be applied to motivic settings. Another point of intersection is chromatic homotopy theory.
Given your background in algebraic geometry, I think it's best to start with motivic homotopy theory and layer in DAG as your comfort with category theory and homotopical algebra grows. Motivic homotopy theory starts with the stuff you already know regarding sites and Grothendieck topologies, and the development from there is partially motivated by classic algebraic topology and homotopy theory, as you've already studied.
To understand either motivic homotopy theory or DAG, you need some familiarity with simplicial sets, so I'd start there. Great references include:
With knowledge of simplicial sets, simplicial presheaves, and the algebraic geometry you know, you can understand motivic spaces. A reference that I love for the homotopy theory of schemes (but that's because it's aimed at an audience with a solid grasp of abstract homotopy theory) is Dugger's manuscript Sheaves and Homotopy.
Next up is motivic spectra. Follow the other roadmap to learn the basics of spectra, generalized cohomology theories, and representability theorems. I think the roadmap to motivic homotopy given over at MSE is solid. Reading through those references, you'd see the motivic lens on algebraic $K$ theory, and how Voevodsky used that connection in his work on the Milnor Conjecture.
As you read the motivic (or even the classical spectra) material, you can't help but come across model categories. They are infused throughout the motivic world and DAG, but I'd recommend to pick them up as you go given your interests, rather than sit down and read the classic references like Dwyer-Spalinski, Hovey, and Hirschhorn (none of which mention motivic things at all).
Going down this path you'll get more used to homotopical algebra and simplicial things, and then it'll be natural to move into higher categories if you find you're still interested to do so. When that time comes, you can choose between Toen et al's Homotopical Algebraic Geometry (HAG) or Lurie's DAG. The best references for the latter remain Lurie's books: Higher Topos Theory, Higher Algebra, and now Spectral Algebraic Geometry.
If you want to see an approach to motivic homotopy theory steeped in the techniques of DAG, then I recommend writings by Marc Hoyois. Nowadays there are plenty of people using $\infty$-categorical tools in motivic settings, so you could get to the point that blends the two areas you said you're interested in.