# Derived algebraic geometry: how to reach research level math?

I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.

My goal is to study derived algebraic geometry, where derived schemes are built out of simplicial commutative rings rather than ordinary commutative rings as in algebraic geometry (there's also a variant using commutative ring spectra, which I don't know anything about). Anyways, since the category of simplicial rings form a model category, we can apply homotopy theoretic methods to study derived schemes.

I thought the first thing I should do is study simplicial homotopy theory, in order to learn about model categories and simplicial objects. So I started reading Simplicial Homotopy Theory by Goerss and Jardine. How should I study this book? There are very few exercises, unlike standard graduate textbooks like Hartshorne, and a lot of the proofs are simplex/diagram chasing, so I decided to skip a lot of the proofs and read the book casually.

A big disadvantage to this method is that I don't understand anything at a deep level and I'm only familiar with a few buzzwords. But I feel overwhelmed by the amount of prerequisite material I need to understand to learn DAG, because most of it is written in the language of $\infty$-categories. So what should I do? How can I get to "research level mathematics"?

EDIT: I'm a senior math major and I've taken the graduate algebraic geometry and algebraic topology sequences. I've also studied some deformation theory.

• A general tip is to read a lot of articles. At first, there are many things that are unclear, but it is similar to being immersed in a language. Eventually, pieces falls into places. – Per Alexandersson Sep 8 '15 at 23:40
• I think this is a fine subject to try and learn. I would question, however, why you have picked this subject if you do not have the requisite background. I would think that you would try to learn this stuff once it is clearly useful and interesting. But if you only know "a few buzzwords", I do not see how it could be interesting. In other words, if you are mostly attracted to this because it seems fancy, you may not have a good time learning it. I would suggest, rather, naturally evolving from the things you already know well and find interesting. – Steven Gubkin Sep 8 '15 at 23:59
• I am not a topologist, but you might want to invest in learning more classical homotopy theory first if you have not already. Then you could look at this mathoverflow.net/questions/132139/…. Following that, if you have a book without exercises, you need to make your own, and ideally you should be talking with other people about the content. Maybe organize a working group? – Steven Gubkin Sep 9 '15 at 0:02
• Goerss and Jardine is a hard book: they leave proofs to the reader that aren't obvious. Try Kamps and Porter's "Abstract Homotopy and Simple Homotopy Theory", which is more classical, but still very abstract. For a free resource, try ncatlab.org/nlab/files/Abstract-Homotopy.pdf – David Roberts Sep 9 '15 at 3:05
• I recommend that before diving into Goerss and Jardine you read the survey article by Greg Friedman to get an intuition for simplicial sets: arxiv.org/PS_cache/arxiv/pdf/0809/0809.4221v3.pdf – Mark Grant Sep 9 '15 at 7:35

I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lurie's Higher algebra.) The plan is based on what worked best for myself, and it's certainly possible that you may prefer to jump into Higher Topos Theory as Yonatan suggested.

0) First of all, make sure you have a solid grounding in basic category theory. For this, read the first two chapters of the excellent lecture notes of Schapira. I would strongly recommend reading chapters 3 and 4 as well, but these can be skipped for now.

Then read chapters I and II of Gabriel-Zisman, Calculus of fractions and homotopy theory, to learn about the theory of localization of categories.

1) The next step is to learn the basics of abstract homotopy theory.

I recommend working through Cisinski's notes. This will take you through simplicial sets, model categories, a beautiful construction of the Quillen and Joyal model structures (which present $\infty$-groupoids and $\infty$-categories, respectively), and the fundamental constructions of $\infty$-category theory (functor categories, homotopy (co)limits, fibred categories, prestacks, etc.).

Supplement the section "Catégories de modèles" with chapter I of Quillen's lecture notes Homotopical algebra.

Then read about stable $\infty$-categories and symmetric monoidal $\infty$-categories in these notes from a mini-course by Cisinski. (By the way, these ones are in English and also summarize very briefly some of the material from the longer course notes). These notes are very brief, so you will have to supplement them with the notes of Joyal. It may also be helpful to have a look at the first chapter of Lurie's Higher algebra and the notes of Moritz Groth.

2) At this point you are ready to learn some derived commutative algebra:

Read lecture 4 of part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry together with section 3 of Lurie's thesis. Supplement this with section 2.2.2 of Toen-Vezzosi's HAG II, referring to chapter 1.2 when necessary. This material is at the heart of derived algebraic geometry: the cotangent complex, infinitesimal extensions, Postnikov towers of simplicial commutative rings, etc.

Other helpful things to look at are Schwede's Diplomarbeit and Quillen's Homology of commutative rings.

3) Before learning about derived stacks, I would strongly recommend working through these notes of Toen about classical algebraic stacks, from a homotopy theoretic perspective. There are also these notes of Preygel. This will make it a lot easier to understand what comes next.

Then, read Lurie's On $\infty$-topoi. It will be helpful to consult sections 15-20 of Cisinski's Bourbaki talk, section 40 of Joyal's notes on quasi-categories, and Rezk's notes. For a summary of this material, see lecture 2 of Moerdijk-Toen.

4) Finally, read about derived stacks in lecture 5 of Moerdijk-Toen and section 5 of Lurie's thesis. Again, chapters 1.3, 1.4, and 2.2 of HAG II will be very helpful references. See also Gaitsgory's notes (he works with commutative connective dg-algebras instead of simplicial commutative rings, but this makes little difference). His notes on quasi-coherent sheaves in DAG are also very good.

5) At this point, you know the definitions of objects in derived algebraic geometry. To get some experience working with them, I would recommend reading some of the following papers:

• Antieau-Gepner, Brauer groups and étale cohomology in derived algebraic geometry, arXiv:1210.0290
• Bhatt, p-adic derived de Rham cohomology, arXiv:1204.6560.
• Bhatt-Scholze, Projectivity of the Witt vector affine Grassmannian, arXiv:1507.06490.
• Gaitsgory-Rozenblyum, A study in derived algebraic geometry, link
• Kerz-Strunk-Tamme, Algebraic K-theory and descent for blow-ups, arXiv:1611.08466.
• Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.
• Toen, Proper lci morphisms preserve perfect complexes, arXiv:1210.2827.
• Toen-Vaquie, Moduli of objects in dg-categories, arXiv:math/0503269.
• The dropbox link for the the second set of notes by Cisinski is broken. – user00000 Jan 31 '17 at 10:12
• Cisinski has written a book "Higher categories and homotopical algebra" based on some of the notes linked to above: a preliminary version is here mathematik.uni-regensburg.de/cisinski/CatLR.pdf – j.c. Apr 27 '18 at 15:54
• this is such a detailed plan, and honestly so helpful. if only there were such a guide for all topics! – Tim Oct 11 '18 at 14:11

In my opinion the best foundations to any modern topic in homotopy theory, and derived algebraic geometry in particular, is "Higher topos theory" of Lurie. The scope covers all the required ($\infty$-)categorical framework, and every chapter starts with a very conceptual motivation. In addition, the book also contains appendices which explain classical material (such as model categories) in a very readable way. You might find in the beginning some proofs which involve technical combinatorics of simplices. Don't be discouraged. Feeling comfortable with simplices is essential and this requires working out some details. The proofs in the book do become increasingly conceptual with each chapter, as the concepts themselves get built and acquire depth.

• Higher Algebra is also an important resource to the algebra of stable homotopy theory and derived algebraic geometry. – user62675 Sep 13 '15 at 22:53
• This is defninetly not enough. The local theory is basically understanding spectra (stable stuff), simplicial rings and dg stuff. And none of these topics is contained in HTT. – user40276 Sep 13 '15 at 23:36
• I agree, HTT is just a good place to start. After that one should delve into a more specific topic. For DAG which is based on spectra, Higher Algebra is undoubtedly the next step. – Yonatan Harpaz Sep 14 '15 at 6:58
• Well, I don't think the entire HTT is strictly necessary. The chapters 1, 5 and 6 seems enough. – user40276 Sep 14 '15 at 18:12
• @010110111 You may refer to the other chapters when necessary. However if you want to get the big picture, 1, 5 and 6 seems enough. After this it's possible to get a good idea of what's an $(\infty, 1)$-stack. Then (if you already have some background in moduli and some intersection theory) you can read the series of Töen and Vezzosi's papers about Homotopical Algebraic Geometry in arxiv and arxiv.org/abs/1401.1044 for the motivations (to use derived $(\infty, 1)$ Artin stacks). Together with these ones you can try Lurie's thesis (however it has few proofs). – user40276 Sep 14 '15 at 23:53

I'm going to take a dissenting view, here. I think the best way to assimilate concepts in derived algebraic geometry (for finite fields, $$\mathbb{R}$$ or $$\mathbb{C}$$), is to understand where and why they are used. Then, work backwards when the need arises. Personally, I found it formidable to read through any section of Toen-Vezzosi's homotopical algebraic geometry series straight through. I'd first recommend reading and understanding the content of Vezzosi's AMS notice, here: https://www.ams.org/notices/201107/rtx110700955p.pdf. Once you begin digesting the need for replacing the source category for Grothendieck's functor of points approach to algebraic geometry with derived commutative algebras, browse through the literature and find instances where this becomes necessary. From my perspective, the most striking application is here: https://arxiv.org/pdf/1102.1150.pdf, where one sees (sloppily speaking here), that even replacing the source category with truncated derived objects goes a very long way in recovering classical results. Feel free to let me know if you'd like me to explicate further.