# Diagram spectra and Algebraic Geometry

I was recently reading a paper titled "Model Categories of Diagram Spectra" and it was mentioned in the paper that the contents of the paper were also useful in algebraic geometry. I'm really interested in this subject and I'm vaguely aware that techniques from stable homotopy are finding ways into algebraic geometry. I'm relatively unfamiliar with Lurie's works on Derived Algebraic Geometry and stuff (which is where I'm guessing a lot of this stuff is used?) so I guess I'm wondering if anyone can suggest some beginner level literature on this subject. Also if anyone is willing to briefly sketch some of the ideas and applications of this too that would be awesome. I understand this question is kind of vague and open ended so maybe it should be a community wiki? Thanks!

Edit: to be more clear, I am particularly interested in a comment made in the paper on ph446 "examples of such symmetric monoidal categories arise in other fields, such as algebraic geometry." I would like to know what this is referring to.

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Could you make your question more focused (as suggested in the FAQ)? –  S. Carnahan May 8 '13 at 1:20
I think the point of the paper is to show that different models of spectra are equivalent. I have no idea what they meant about the algebraic geometry, but it probably wasn't Lurie's work since the paper came out in 1999. –  Sean Tilson May 8 '13 at 4:21
My guess is that this is related to the Asterisque volume by May and Kriz that talks about motives, but that is just a guess. There's a version of the volume here: math.uiuc.edu/K-theory/0061 –  Dan Ramras May 8 '13 at 6:19
I'd start with Morel-Voevodsky's $\mathbb A^1$-homotopy theory. –  Fernando Muro May 8 '13 at 6:46
I've not had time to answer, but yes, we were thinking mostly of motivic homotopy theory, as David and Fernando surmise. Dan, that kind of motives (mixed Tate motives) was not what we were thinking of, although I have some faint ideas of a relationship. –  Peter May May 9 '13 at 3:52

A nice paper to understand the connection between Spectra and Motivic Homotopy Theory is Mark Hovey's Spectra and symmetric spectra in general model categories. To form the classical category of spectra you start with Topological Spaces and an endofunctor $\Sigma$ which you wish to stabilize. The trick is to pass to sequences of spaces $(X_i)$ with structure maps $\Sigma X_i \to X_{i+1}$. To do symmetric spectra you build in group actions as you go. In this paper Hovey discussed how to do this in a more general model category $M$ and with a general endofunctor $G$. With enough hypotheses you can prove things like "the stable equivalences are the stable homotopy isomorphisms"

The work of Morel and Voevodsky was the starting place for motivic homotopy theory. They resolved the Milnor conjecture. Model categories came into the picture because they wanted a model structure on the category of schemes. This doesn't quite work, but if you fatten up the category by formally throwing in the missing colimits then it does work. The other trick is to use the Nisnevich site. The punchline is that you now have a category where both topological objects and algebraic objects live. We've done this in a way such that the topological spheres (well, technically simplicial) and the affine line $\mathbb{A}^1$ are spheres, i.e. monoidal units. If you want more details on this work I can add some but I'm going to assume you can go read about it on your own and I'm going to focus on where spectra come in instead.

In the introduction to the paper above, Hovey discusses the application to Morel-Voevodsky. There are several applications. For one, if you want to do spectra in the motivic setting you have to change what you mean by suspension because now there are different types of spheres, as mentioned above. So Hovey's work lets you do this, i.e. pass from unstable $\mathbb{A}^1$-homotopy theory to stable $\mathbb{A}^1$-homotopy theory. Jardine had proven stable equivalences are stable homotopy isomorphisms in the stable $\mathbb{A}^1$-homotopy theory using the Nisnevitch descent theorem. With Hovey's machine (and passage to a nicer model category which is equivalent to the Morel-Voevodsky one) gets this result in a cleaner way without relying on that theorem.

Motivic homotopy theory has become a thriving subject. Jardine has written about motivic symmetric spectra, Po Hu has written about motivic S-modules, and Ostvaer has written about motivic functors (which might be to your liking if you like the Diagram Spectra paper). Googling will give you these papers. I recommend checking out the presentations on Kyle Ormsby's website for an accessible introduction to the field.

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If you want to know more about the applications of Motivic Homotopy Theory to number theory and algebraic geometry, check out my recent question and the marvellous answer I got: mathoverflow.net/questions/129762/… –  David White May 9 '13 at 13:34
@David, if you like it, why don't you accept it? –  Fernando Muro May 9 '13 at 16:04
I didn't realize that I hadn't. I have now remedied the problem. I probably didn't accept immediately because I was hoping for more answers because my question was so broad. –  David White May 9 '13 at 17:39
This is awesome! Thanks for the references, I will check them out –  Geoffrey May 9 '13 at 23:34