# Exposition of Grothendieck's mathematics

As Wikipedia says:

In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions which he believed qualified as "great ideas". In chronological order, they are:

1. Topological tensor products and nuclear spaces.
2. "Continuous" and "discrete" duality (derived categories, "six operations").
3. Yoga of the Grothendieck–Riemann–Roch theorem (K-theory, relation with intersection theory).
4. Schemes.
5. Topoi.
6. Étale cohomology and l-adic cohomology.
7. Motives and the motivic Galois group (Grothendieck ⊗-categories).
8. Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients", ...
9. "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra.
10. Tame topology.
11. Yoga of anabelian algebraic geometry, Galois–Teichmüller theory.
12. "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds.

What are some modern, concise expository texts on these topics that students can use to learn the great ideas of Grothendieck? EGA, SGA, Les Dérivateurs, La Longue Marche, and so on don't count, because they are French, overwhelming, and maybe a bit outdated (I'm not sure) - anyway, it's not realistic for a student to go through them in their free time in addition to the courses they take for their degree. If you want to give a textbook, make sure it's as short as possible.

Let me start by giving two examples of the kind of answers I am expecting:

• May I suggest to include the two examples into (separate) answers just to know (as someone else suggested) the relative interests of MathOverflow audience... thanks! Feb 3, 2022 at 2:04
• Apocryphally Gauss said, "There is no royal road to mathematics." There is also realistically no way to learn all that someone has to say by reading other people's expositions! Finally, a lot of good mathematics is written in French. Learn to read it. Seriously! Do not be put off by the length. After some time you learn to read faster. Feb 3, 2022 at 3:20
• @Kapil I only agree if you give this piece of advice to somehow who is really sure he wants to do research in algebraic geometry. Then the EGA's and SGA's may be better then modern and concise expositions. Feb 3, 2022 at 13:19
• because they are French - this does contradict your modern, concise, for students conditions. If you want to specify in English, just specify that. Otherwise, is Italian or Japanese okay? Feb 3, 2022 at 14:07
• @AlexanderWoo Everybody decides for themselves what is a waste of time for them. If someone enjoys learning algebraic geometry without doing research in it, why not. Feb 3, 2022 at 21:58

A good roadmap for FGA (topic 4, with glimpses on topic 6) is

Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo, Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3541-6/hbk). x, 339 p. (2005). ZBL1085.14001.

• Thanks! Please give one of the topics 1-12 this gets associated to. Feb 2, 2022 at 14:34

For topic 4 (schemes), I'd suggest Schemes: The Language of Algebraic Geometry, by Eisenbud and Harris. This book is out of print, and is generally regarded as having been "superseded" by its successor, The Geometry of Schemes. But for your purposes, I think the earlier book is actually better, since it's only 160 pages, and if you just want the main ideas, the first two chapters should suffice. The authors specifically aimed at making the topic accessible to students encountering schemes for the first time.

• I believe the survey in Mumford's "Lectures on curves in an algebraic surface" is even shorter (and, of course, it is written by Mumford). Feb 3, 2022 at 0:50
• I've always felt a bit self-conscious about schemes, because in spite of having taken a course or two back in the day I don't feel like I could direct a short infomercial on why schemes are important - this is in contrast to, say, manifolds or Hilbert spaces, where I am more confident I could put together a good sales pitch. Would Eisenbud-Harris' or Mumford's books help with this? Feb 3, 2022 at 0:56
• @PaulSiegel I do think that Eisenbud-Harris (Chapter 2 especially) would help you. But for an elevator pitch, maybe the short section on schemes by Danilov in Algebraic Curves, Algebraic Manifolds and Schemes is even more direct. See also this math.SE question. Feb 3, 2022 at 2:58
• Mumford’s book is precisely about presenting Grothendieck’s proof, via schemes, of the “Completeness Theorem” that led to such bitter fight between Severi and Enriques. Feb 3, 2022 at 3:33

Let me suggest having looking at Grothendieck's own article The cohomology theory of abstract algebraic varieties in the proceedings of the 1958 ICM. This is both short (15 pages) and in English. Although written quite early on, it explains the motivations and intuitions for many of the developments that lay in the future in a lucid way. For instance, he compares and contrast his notion of scheme (referred to here as a pre-schema) with the more classical notions. He hints at "a definition of the Weil cohomology (involving both 'spatial' and Galois cohomology)...", which would eventually become étale cohomology.

• this is a wonderful suggestion. In addition, although I may be the only person with a copy, Dieudonne's 1962 Maryland notes, formerly available from Harvard math dept, are a summary of EGA at about 105 pages. I myself however strongly recommend reading at least selected pages of the longer versions, EGA, SGA, etc as absolutely unmatched treatments written by the original creators of these ideas. You will be very glad of every word you read by Grothendieck, or Mumford. May 12, 2022 at 2:37

For topic 3, I would say Hartshorne's Appendix A is pretty good as a VERY brief introduction. After that, as also referred to there, I would recommend Manin's Lectures on the K-functor in algebraic geometry.

For topic 5, try Leinster's nice expository article https://arxiv.org/abs/1012.5647.

Topic 10 got developed further via the concept of o-minimality. See e.g. "Tame topology and o-minimal structures" by Lou van den Dries, 1998 CUP, http://dx.doi.org/10.1017/CBO9780511525919

A shorter text is by Michel Coste: http://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf

• Note the intersection of Coste's mentioned text with his book Real Algebraic Geometry, by Jacek Bochnak, Michel Coste, and Marie-Francoise Roy. Feb 4, 2022 at 6:30

For learning about topoi, I'd strongly recommend Goldblatt's Topoi: The Categorical Analysis of Logic, which goes over the subject and a few interesting applications in a very approachable and pleasant manner. It's not particularly concise at 556 pages, though you needn't read the entire text to get a solid foundational concept of topoi.

For topic 7, Milne's article https://www.jmilne.org/math/xnotes/MOT.pdf could be useful.

For topic 6, there are Milne's notes: https://www.jmilne.org/math/CourseNotes/LEC.pdf

In the front matter Milne writes:

These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes.

1. Infinity-stacks

Before learning about infinity stacks, one ought to learn about stacks. For this, look at section 4.1.1 of Vistoli's notes, Notes on Grothendieck topologies, fibred categories and descent theory, which is an exposition of one concrete case. This section is around a page and a half. He states:

that the fact we can glue continuous maps and topological spaces says $$(Cont)$$ is a stack over $$Top$$.

This in fact a generalisation of the clutching construction of bundles. And you can learn about bundles in the very readable first chapter of Grothendiecks Kansas notes on fibre bundles.

There is also an expository note by Barbara Fantechi, Stacks for Everybody.

• Thanks! Concerning topic 9, I wonder what he means by "cohomological formalism of topoi as inspiration for a new homotopical algebra" - what have topoi to do with homotopical algebra? Could he mean $\infty$-topoi à la Lurie? Feb 3, 2022 at 15:08
• @user476368: There is a duality between bundles and sheafs. Likewise with higher versions of these. A 2-sheaf is exactly a stack. According to Nlab, Quillen showed how to do abstract homotopy via model categories and this was called homotopical algebra. Model categories are a presentation of infinity topoi which suggests homotopical algebra can be generalised to this setting. Feb 3, 2022 at 15:25
• @MoziburUllah Model categories do not (necessarily) present ∞-topoi, only those called model topoi do. In general model categories present complete and cocomplete ∞-categories (exactly which of those is not fully known, IIRC - certainly the presentable ones though) Feb 3, 2022 at 16:16

For topic 1, the book by Diestel, Fourie, Swart: https://books.google.de/books?id=VBg5cGSngRoC&printsec=copyright&redir_esc=y#v=onepage&q&f=false provides a good introduction.

For 5. Topoi. There is a very readable notice of the AMS: What is... a topos?, Luc Illusie, and with M Raynaud about Schemes at Grothendieck and Algebraic Geometry. In fact, the webpage of Professor L. Illusie contains very valuable material regarding the question.

You can read (online) about various of the listed topics in Lectures grothendieckiennes Edited by Frédéric Jaëck

And find a lot of original material at Thèmes pour une Harmonie!

On topic 1, I'd like to recommend Ray Ryan's Introduction to tensor products of Banach spaces, Springer 2002. Zbl 1090.46001